Misplaced Pages

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145-848: By technical definition, it is a method of constructing an orthonormal basis from a set of vectors in an inner product space , most commonly the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} equipped with the standard inner product . The Gram–Schmidt process takes a finite , linearly independent set of vectors S = { v 1 , … , v k } {\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}} for k ≤ n and generates an orthogonal set S ′ = { u 1 , … , u k } {\displaystyle S'=\{\mathbf {u} _{1},\ldots ,\mathbf {u} _{k}\}} that spans

290-442: A ‖ 2 = 1 {\displaystyle \langle e_{i},e_{i}\rangle =\|e_{a}\|^{2}=1} for each index i . {\displaystyle i.} This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let V {\displaystyle V} be any inner product space. Then a collection E = { e

435-502: A , e a ⟩ = ‖ e a ‖ 2 = 1 {\displaystyle \langle e_{a},e_{a}\rangle =\|e_{a}\|^{2}=1} for all a , b ∈ A . {\displaystyle a,b\in A.} Using an infinite-dimensional analog of the Gram-Schmidt process one may show: Theorem. Any separable inner product space has an orthonormal basis. Using

580-676: A b b d ] [ y 1 y 2 ] = a x 1 y 1 + b x 1 y 2 + b x 2 y 1 + d x 2 y 2 . {\displaystyle \langle x,y\rangle :=x^{\operatorname {T} }\mathbf {M} y=\left[x_{1},x_{2}\right]{\begin{bmatrix}a&b\\b&d\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}y_{2}.} As mentioned earlier, every inner product on R 2 {\displaystyle \mathbb {R} ^{2}}

725-461: A } a ∈ A {\displaystyle E=\left\{e_{a}\right\}_{a\in A}} is a basis for V {\displaystyle V} if the subspace of V {\displaystyle V} generated by finite linear combinations of elements of E {\displaystyle E} is dense in V {\displaystyle V} (in the norm induced by

870-764: A 1 , b 1 , … , a n , b n ) ∈ R 2 n {\displaystyle \left(a_{1},b_{1},\ldots ,a_{n},b_{n}\right)\in \mathbb {R} ^{2n}} ), then the dot product x ⋅ y = ( x 1 , … , x 2 n ) ⋅ ( y 1 , … , y 2 n ) := x 1 y 1 + ⋯ + x 2 n y 2 n {\displaystyle x\,\cdot \,y=\left(x_{1},\ldots ,x_{2n}\right)\,\cdot \,\left(y_{1},\ldots ,y_{2n}\right):=x_{1}y_{1}+\cdots +x_{2n}y_{2n}} defines

1015-608: A , b ∈ F {\displaystyle a,b\in F} . If the positive-definiteness condition is replaced by merely requiring that ⟨ x , x ⟩ ≥ 0 {\displaystyle \langle x,x\rangle \geq 0} for all x {\displaystyle x} , then one obtains the definition of positive semi-definite Hermitian form . A positive semi-definite Hermitian form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle }

1160-411: A , b ] . {\displaystyle [a,b].} The inner product is ⟨ f , g ⟩ = ∫ a b f ( t ) g ( t ) ¯ d t . {\displaystyle \langle f,g\rangle =\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.} This space is not complete; consider for example, for

1305-441: A Hamel basis E ∪ F {\displaystyle E\cup F} for K , {\displaystyle K,} where E ∩ F = ∅ . {\displaystyle E\cap F=\varnothing .} Since it is known that the Hamel dimension of K {\displaystyle K} is c , {\displaystyle c,}

1450-465: A and b are arbitrary scalars. Over R {\displaystyle \mathbb {R} } , conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form . The binomial expansion of a square becomes Some authors, especially in physics and matrix algebra , prefer to define inner products and sesquilinear forms with linearity in

1595-501: A complex vector space with an operation called an inner product . The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in ⟨ a , b ⟩ {\displaystyle \langle a,b\rangle } . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles , and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces , in which

SECTION 10

#1733085140553

1740-467: A symmetric positive-definite matrix M {\displaystyle \mathbf {M} } such that ⟨ x , y ⟩ = x T M y {\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y} for all x , y ∈ R n . {\displaystyle x,y\in \mathbb {R} ^{n}.} If M {\displaystyle \mathbf {M} }

1885-535: A (algebraic) basis of a Hilbert space (or, more generally, a basis of any dense subspace), it yields a (functional-analytic) orthonormal basis. Note that in the general case often the strict inequality κ < λ {\displaystyle \kappa <\lambda } holds, even if the starting set was linearly independent, and the span of ( u α ) α < κ {\displaystyle (u_{\alpha })_{\alpha <\kappa }} need not be

2030-505: A Hilbert space H {\displaystyle H} is called an orthonormal system. An orthonormal basis is an orthonormal system with the additional property that the linear span of S {\displaystyle S} is dense in H {\displaystyle H} . Alternatively, the set S {\displaystyle S} can be regarded as either complete or incomplete with respect to H {\displaystyle H} . That is, we can take

2175-746: A bijection. Then there is a linear transformation T : K → L {\displaystyle T:K\to L} such that T f = φ ( f ) {\displaystyle Tf=\varphi (f)} for f ∈ F , {\displaystyle f\in F,} and T e = 0 {\displaystyle Te=0} for e ∈ E . {\displaystyle e\in E.} Let V = K ⊕ L {\displaystyle V=K\oplus L} and let G = { ( k , T k ) : k ∈ K } {\displaystyle G=\{(k,Tk):k\in K\}} be

2320-501: A computer, the vectors u k {\displaystyle \mathbf {u} _{k}} are often not quite orthogonal, due to rounding errors . For the Gram–Schmidt process as described above (sometimes referred to as "classical Gram–Schmidt") this loss of orthogonality is particularly bad; therefore, it is said that the (classical) Gram–Schmidt process is numerically unstable . The Gram–Schmidt process can be stabilized by

2465-401: A general inner product space V , {\displaystyle V,} an orthonormal basis can be used to define normalized orthogonal coordinates on V . {\displaystyle V.} Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to

2610-499: A non-degenerate symmetric bilinear form known as the metric tensor . In such a basis, the metric takes the form diag ( + 1 , ⋯ , + 1 , − 1 , ⋯ , − 1 ) {\displaystyle {\text{diag}}(+1,\cdots ,+1,-1,\cdots ,-1)} with p {\displaystyle p} positive ones and q {\displaystyle q} negative ones. If B {\displaystyle B}

2755-750: A non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references). Let K {\displaystyle K} be a Hilbert space of dimension ℵ 0 . {\displaystyle \aleph _{0}.} (for instance, K = ℓ 2 ( N ) {\displaystyle K=\ell ^{2}(\mathbb {N} )} ). Let E {\displaystyle E} be an orthonormal basis of K , {\displaystyle K,} so | E | = ℵ 0 . {\displaystyle |E|=\aleph _{0}.} Extend E {\displaystyle E} to

2900-399: A parametrized version of the Gram–Schmidt process yields a (strong) deformation retraction of the general linear group G L ( R n ) {\displaystyle \mathrm {GL} (\mathbb {R} ^{n})} onto the orthogonal group O ( R n ) {\displaystyle O(\mathbb {R} ^{n})} . When this process is implemented on

3045-433: A popular and effective algorithm for even the largest electronic structure calculations. Gram-Schmidt orthogonalization can be done in strongly-polynomial time . The run-time analysis is similar to that of Gaussian elimination . Orthonormal basis In mathematics , particularly linear algebra , an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension

SECTION 20

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3190-963: A positive definite symmetric bilinear form ϕ = ⟨ ⋅ , ⋅ ⟩ {\displaystyle \phi =\langle \cdot ,\cdot \rangle } . One way to view an orthonormal basis with respect to ϕ {\displaystyle \phi } is as a set of vectors B = { e i } {\displaystyle {\mathcal {B}}=\{e_{i}\}} , which allow us to write v = v i e i     ∀   v ∈ V {\displaystyle v=v^{i}e_{i}\ \ \forall \ v\in V} , and v i ∈ R {\displaystyle v^{i}\in \mathbb {R} } or ( v i ) ∈ R n {\displaystyle (v^{i})\in \mathbb {R} ^{n}} . With respect to this basis,

3335-517: A pre-Hilbert space H , {\displaystyle H,} an orthonormal basis for H {\displaystyle H} is an orthonormal set of vectors with the property that every vector in H {\displaystyle H} can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for H . {\displaystyle H.} Note that an orthonormal basis in this sense

3480-523: A real inner product on the real vector space V R . {\displaystyle V_{\mathbb {R} }.} Every inner product on a real vector space is a bilinear and symmetric map . For example, if V = C {\displaystyle V=\mathbb {C} } with inner product ⟨ x , y ⟩ = x y ¯ , {\displaystyle \langle x,y\rangle =x{\overline {y}},} where V {\displaystyle V}

3625-1001: A real inner product on this space. The unique complex inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } on V = C n {\displaystyle V=\mathbb {C} ^{n}} induced by the dot product is the map that sends c = ( c 1 , … , c n ) , d = ( d 1 , … , d n ) ∈ C n {\displaystyle c=\left(c_{1},\ldots ,c_{n}\right),d=\left(d_{1},\ldots ,d_{n}\right)\in \mathbb {C} ^{n}} to ⟨ c , d ⟩ := c 1 d 1 ¯ + ⋯ + c n d n ¯ {\displaystyle \langle c,d\rangle :=c_{1}{\overline {d_{1}}}+\cdots +c_{n}{\overline {d_{n}}}} (because

3770-935: A small modification; this version is sometimes referred to as modified Gram-Schmidt or MGS. This approach gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic. Instead of computing the vector u k as u k = v k − proj u 1 ⁡ ( v k ) − proj u 2 ⁡ ( v k ) − ⋯ − proj u k − 1 ⁡ ( v k ) , {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{k})-\cdots -\operatorname {proj} _{\mathbf {u} _{k-1}}(\mathbf {v} _{k}),} it

3915-2004: A subspace of the span of ( v α ) α < λ {\displaystyle (v_{\alpha })_{\alpha <\lambda }} (rather, it's a subspace of its completion). Consider the following set of vectors in R 2 {\displaystyle \mathbb {R} ^{2}} (with the conventional inner product ) S = { v 1 = [ 3 1 ] , v 2 = [ 2 2 ] } . {\displaystyle S=\left\{\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2\\2\end{bmatrix}}\right\}.} Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors: u 1 = v 1 = [ 3 1 ] {\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}}} u 2 = v 2 − proj u 1 ⁡ ( v 2 ) = [ 2 2 ] − proj [ 3 1 ] ⁡ [ 2 2 ] = [ 2 2 ] − 8 10 [ 3 1 ] = [ − 2 / 5 6 / 5 ] . {\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2})={\begin{bmatrix}2\\2\end{bmatrix}}-\operatorname {proj} _{\left[{\begin{smallmatrix}3\\1\end{smallmatrix}}\right]}{\begin{bmatrix}2\\2\end{bmatrix}}={\begin{bmatrix}2\\2\end{bmatrix}}-{\frac {8}{10}}{\begin{bmatrix}3\\1\end{bmatrix}}={\begin{bmatrix}-2/5\\6/5\end{bmatrix}}.} We check that

4060-474: A vector and a covector. Every inner product space induces a norm , called its canonical norm , that is defined by ‖ x ‖ = ⟨ x , x ⟩ . {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.} With this norm, every inner product space becomes a normed vector space . So, every general property of normed vector spaces applies to inner product spaces. In particular, one has

4205-413: A vector space over C {\displaystyle \mathbb {C} } that becomes an inner product space with the inner product ⟨ x , y ⟩ := x y ¯  for  x , y ∈ C . {\displaystyle \langle x,y\rangle :=x{\overline {y}}\quad {\text{ for }}x,y\in \mathbb {C} .} Unlike with

4350-608: Is Hermitian and positive definite , so it can be written as V ∗ V = L L ∗ , {\displaystyle V^{*}V=LL^{*},} using the Cholesky decomposition . The lower triangular matrix L {\displaystyle L} with strictly positive diagonal entries is invertible . Then columns of the matrix U = V ( L − 1 ) ∗ {\displaystyle U=V\left(L^{-1}\right)^{*}} are orthonormal and span

4495-743: Is isomorphic to ℓ 2 ( B ) {\displaystyle \ell ^{2}(B)} in the following sense: there exists a bijective linear map Φ : H → ℓ 2 ( B ) {\displaystyle \Phi :H\to \ell ^{2}(B)} such that ⟨ Φ ( x ) , Φ ( y ) ⟩ = ⟨ x , y ⟩     ∀   x , y ∈ H . {\displaystyle \langle \Phi (x),\Phi (y)\rangle =\langle x,y\rangle \ \ \forall \ x,y\in H.} A set S {\displaystyle S} of mutually orthonormal vectors in

Gram–Schmidt process - Misplaced Pages Continue

4640-505: Is uncountable , only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x , {\displaystyle x,} and the formula is usually known as Parseval's identity . If B {\displaystyle B} is an orthonormal basis of H , {\displaystyle H,} then H {\displaystyle H}

4785-441: Is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function. For real random variables X {\displaystyle X} and Y , {\displaystyle Y,} the expected value of their product ⟨ X , Y ⟩ = E [ X Y ] {\displaystyle \langle X,Y\rangle =\mathbb {E} [XY]}

4930-409: Is a basis for V {\displaystyle V} whose vectors are orthonormal , that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of

5075-504: Is a complete orthonormal set. Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces , with separate cases depending on whether

5220-447: Is a linear subspace of H ¯ , {\displaystyle {\overline {H}},} the inner product of H {\displaystyle H} is the restriction of that of H ¯ , {\displaystyle {\overline {H}},} and H {\displaystyle H} is dense in H ¯ {\displaystyle {\overline {H}}} for

5365-474: Is a continuous linear operator that satisfies ⟨ x , A x ⟩ = 0 {\displaystyle \langle x,Ax\rangle =0} for all x ∈ V , {\displaystyle x\in V,} then A = 0. {\displaystyle A=0.} This statement is no longer true if ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle }

5510-410: Is a linear combination of v 1 , … , v i − 1 {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}} . If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by

5655-548: Is a linear map (linear for both V {\displaystyle V} and V R {\displaystyle V_{\mathbb {R} }} ) that denotes rotation by 90 ∘ {\displaystyle 90^{\circ }} in the plane. Because x {\displaystyle x} and A x {\displaystyle Ax} are perpendicular vectors and ⟨ x , A x ⟩ R {\displaystyle \langle x,Ax\rangle _{\mathbb {R} }}

5800-455: Is a real vector space then ⟨ x , y ⟩ = Re ⁡ ⟨ x , y ⟩ = 1 4 ( ‖ x + y ‖ 2 − ‖ x − y ‖ 2 ) {\displaystyle \langle x,y\rangle =\operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)} and

5945-438: Is a vector space over the field C , {\displaystyle \mathbb {C} ,} then V R = R 2 {\displaystyle V_{\mathbb {R} }=\mathbb {R} ^{2}} is a vector space over R {\displaystyle \mathbb {R} } and ⟨ x , y ⟩ R {\displaystyle \langle x,y\rangle _{\mathbb {R} }}

Gram–Schmidt process - Misplaced Pages Continue

6090-434: Is an inner product if and only if for all x {\displaystyle x} , if ⟨ x , x ⟩ = 0 {\displaystyle \langle x,x\rangle =0} then x = 0 {\displaystyle x=\mathbf {0} } . In the following properties, which result almost immediately from the definition of an inner product, x , y and z are arbitrary vectors, and

6235-428: Is an inner product. In this case, ⟨ X , X ⟩ = 0 {\displaystyle \langle X,X\rangle =0} if and only if P [ X = 0 ] = 1 {\displaystyle \mathbb {P} [X=0]=1} (that is, X = 0 {\displaystyle X=0} almost surely ), where P {\displaystyle \mathbb {P} } denotes

6380-508: Is an isometric linear map V → ℓ 2 {\displaystyle V\rightarrow \ell ^{2}} with a dense image. This theorem can be regarded as an abstract form of Fourier series , in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials . Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided ℓ 2 {\displaystyle \ell ^{2}}

6525-595: Is an orthogonal basis of H , {\displaystyle H,} then every element x ∈ H {\displaystyle x\in H} may be written as x = ∑ b ∈ B ⟨ x , b ⟩ ‖ b ‖ 2 b . {\displaystyle x=\sum _{b\in B}{\frac {\langle x,b\rangle }{\lVert b\rVert ^{2}}}b.} When B {\displaystyle B}

6670-642: Is an orthonormal basis of the space C [ − π , π ] {\displaystyle C[-\pi ,\pi ]} with the L 2 {\displaystyle L^{2}} inner product. The mapping f ↦ 1 2 π { ∫ − π π f ( t ) e − i k t d t } k ∈ Z {\displaystyle f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}

6815-481: Is called the Stiefel manifold V n ( R n ) {\displaystyle V_{n}(\mathbb {R} ^{n})} of orthonormal n {\displaystyle n} -frames . In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, there is no natural choice of orthonormal basis, but once one

6960-530: Is closely related to the expression using determinants above. Other orthogonalization algorithms use Householder transformations or Givens rotations . The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the j {\displaystyle j} th orthogonalized vector after the j {\displaystyle j} th iteration, while orthogonalization using Householder reflections produces all

7105-1967: Is computed as u k ( 1 ) = v k − proj u 1 ⁡ ( v k ) , u k ( 2 ) = u k ( 1 ) − proj u 2 ⁡ ( u k ( 1 ) ) , ⋮ u k ( k − 2 ) = u k ( k − 3 ) − proj u k − 2 ⁡ ( u k ( k − 3 ) ) , u k ( k − 1 ) = u k ( k − 2 ) − proj u k − 1 ⁡ ( u k ( k − 2 ) ) , e k = u k ( k − 1 ) ‖ u k ( k − 1 ) ‖ {\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k}),\\\mathbf {u} _{k}^{(2)}&=\mathbf {u} _{k}^{(1)}-\operatorname {proj} _{\mathbf {u} _{2}}\left(\mathbf {u} _{k}^{(1)}\right),\\&\;\;\vdots \\\mathbf {u} _{k}^{(k-2)}&=\mathbf {u} _{k}^{(k-3)}-\operatorname {proj} _{\mathbf {u} _{k-2}}\left(\mathbf {u} _{k}^{(k-3)}\right),\\\mathbf {u} _{k}^{(k-1)}&=\mathbf {u} _{k}^{(k-2)}-\operatorname {proj} _{\mathbf {u} _{k-1}}\left(\mathbf {u} _{k}^{(k-2)}\right),\\\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}^{(k-1)}}{\left\|\mathbf {u} _{k}^{(k-1)}\right\|}}\end{aligned}}} This method

7250-593: Is considered as a real vector space in the usual way (meaning that it is identified with the 2 n − {\displaystyle 2n-} dimensional real vector space R 2 n , {\displaystyle \mathbb {R} ^{2n},} with each ( a 1 + i b 1 , … , a n + i b n ) ∈ C n {\displaystyle \left(a_{1}+ib_{1},\ldots ,a_{n}+ib_{n}\right)\in \mathbb {C} ^{n}} identified with (

7395-604: Is defined appropriately, as is explained in the article Hilbert space ). In particular, we obtain the following result in the theory of Fourier series: Theorem. Let V {\displaystyle V} be the inner product space C [ − π , π ] . {\displaystyle C[-\pi ,\pi ].} Then the sequence (indexed on set of all integers) of continuous functions e k ( t ) = e i k t 2 π {\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}

SECTION 50

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7540-548: Is defined as proj u ⁡ ( v ) = ⟨ v , u ⟩ ⟨ u , u ⟩ u , {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {v} ,\mathbf {u} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,} where ⟨ v , u ⟩ {\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle } denotes

7685-410: Is denoted 0 {\displaystyle \mathbf {0} } for distinguishing it from the scalar 0 . An inner product space is a vector space V over the field F together with an inner product , that is, a map that satisfies the following three properties for all vectors x , y , z ∈ V {\displaystyle x,y,z\in V} and all scalars

7830-985: Is dense in V . {\displaystyle V.} Finally, { ( e , 0 ) : e ∈ E } {\displaystyle \{(e,0):e\in E\}} is a maximal orthonormal set in G {\displaystyle G} ; if 0 = ⟨ ( e , 0 ) , ( k , T k ) ⟩ = ⟨ e , k ⟩ + ⟨ 0 , T k ⟩ = ⟨ e , k ⟩ {\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle } for all e ∈ E {\displaystyle e\in E} then k = 0 , {\displaystyle k=0,} so ( k , T k ) = ( 0 , 0 ) {\displaystyle (k,Tk)=(0,0)}

7975-761: Is equivalent to the expression using the proj {\displaystyle \operatorname {proj} } operator defined above. The results can equivalently be expressed as u k = v k ∧ v k − 1 ∧ ⋅ ⋅ ⋅ ∧ v 1 ( v k − 1 ∧ ⋅ ⋅ ⋅ ∧ v 1 ) − 1 , {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}\wedge \mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1}(\mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1})^{-1},} which

8120-509: Is generalized by the Iwasawa decomposition . The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix ). The vector projection of a vector v {\displaystyle \mathbf {v} } on a nonzero vector u {\displaystyle \mathbf {u} }

8265-655: Is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis. The other Stiefel manifolds V k ( R n ) {\displaystyle V_{k}(\mathbb {R} ^{n})} for k < n {\displaystyle k<n} of incomplete orthonormal bases (orthonormal k {\displaystyle k} -frames) are still homogeneous spaces for

8410-508: Is instead a real inner product, as this next example shows. Suppose that V = C {\displaystyle V=\mathbb {C} } has the inner product ⟨ x , y ⟩ := x y ¯ {\displaystyle \langle x,y\rangle :=x{\overline {y}}} mentioned above. Then the map A : V → V {\displaystyle A:V\to V} defined by A x = i x {\displaystyle Ax=ix}

8555-416: Is just the dot product, ⟨ x , A x ⟩ R = 0 {\displaystyle \langle x,Ax\rangle _{\mathbb {R} }=0} for all vectors x ; {\displaystyle x;} nevertheless, this rotation map A {\displaystyle A} is certainly not identically 0. {\displaystyle 0.} In contrast, using

8700-535: Is known as Gram–Schmidt orthogonalization , and the calculation of the sequence e 1 , … , e k {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}} is known as Gram–Schmidt orthonormalization . To check that these formulas yield an orthogonal sequence, first compute ⟨ u 1 , u 2 ⟩ {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle } by substituting

8845-557: Is known as the Hermitian form and is given by ⟨ x , y ⟩ = y † M x = x † M y ¯ , {\displaystyle \langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},} where M {\displaystyle M} is any Hermitian positive-definite matrix and y † {\displaystyle y^{\dagger }}

SECTION 60

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8990-574: Is mainly of theoretical interest. Expressed using notation used in geometric algebra , the unnormalized results of the Gram–Schmidt process can be expressed as u k = v k − ∑ j = 1 k − 1 ( v k ⋅ u j ) u j − 1   , {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}(\mathbf {v} _{k}\cdot \mathbf {u} _{j})\mathbf {u} _{j}^{-1}\ ,} which

9135-579: Is not defined in V R , {\displaystyle V_{\mathbb {R} },} the vector in V {\displaystyle V} denoted by i x {\displaystyle ix} is nevertheless still also an element of V R {\displaystyle V_{\mathbb {R} }} ). For the complex inner product, ⟨ x , i x ⟩ = − i ‖ x ‖ 2 , {\displaystyle \langle x,ix\rangle =-i\|x\|^{2},} whereas for

9280-439: Is not generally a Hamel basis , since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H , {\displaystyle H,} although not necessarily the entire space. If we go on to Hilbert spaces , a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on

9425-413: Is of this form (where b ∈ R , a > 0 {\displaystyle b\in \mathbb {R} ,a>0} and d > 0 {\displaystyle d>0} satisfy a d > b 2 {\displaystyle ad>b^{2}} ). The general form of an inner product on C n {\displaystyle \mathbb {C} ^{n}}

9570-715: Is orthonormal, this simplifies to x = ∑ b ∈ B ⟨ x , b ⟩ b {\displaystyle x=\sum _{b\in B}\langle x,b\rangle b} and the square of the norm of x {\displaystyle x} can be given by ‖ x ‖ 2 = ∑ b ∈ B | ⟨ x , b ⟩ | 2 . {\displaystyle \|x\|^{2}=\sum _{b\in B}|\langle x,b\rangle |^{2}.} Even if B {\displaystyle B}

9715-782: Is positive-definite (which happens if and only if det M = a d − b 2 > 0 {\displaystyle \det \mathbf {M} =ad-b^{2}>0} and one/both diagonal elements are positive) then for any x := [ x 1 , x 2 ] T , y := [ y 1 , y 2 ] T ∈ R 2 , {\displaystyle x:=\left[x_{1},x_{2}\right]^{\operatorname {T} },y:=\left[y_{1},y_{2}\right]^{\operatorname {T} }\in \mathbb {R} ^{2},} ⟨ x , y ⟩ := x T M y = [ x 1 , x 2 ] [

9860-434: Is the j {\displaystyle j} th vector) are replaced by orthonormal vectors (columns of U ) which span the same subspace. The cost of this algorithm is asymptotically O( nk ) floating point operations, where n is the dimensionality of the vectors. If the rows { v 1 , ..., v k } are written as a matrix A {\displaystyle A} , then applying Gaussian elimination to

10005-1614: Is the Gram determinant D j = | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v 2 , v 2 ⟩ ⋯ ⟨ v j , v 2 ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v 1 , v j ⟩ ⟨ v 2 , v j ⟩ ⋯ ⟨ v j , v j ⟩ | . {\displaystyle D_{j}={\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j}\rangle \end{vmatrix}}.} Note that

10150-474: Is the conjugate transpose of y . {\displaystyle y.} For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation. The article on Hilbert spaces has several examples of inner product spaces, wherein

10295-604: Is the dot product x ⋅ y , {\displaystyle x\cdot y,} where x = a + i b ∈ V = C {\displaystyle x=a+ib\in V=\mathbb {C} } is identified with the point ( a , b ) ∈ V R = R 2 {\displaystyle (a,b)\in V_{\mathbb {R} }=\mathbb {R} ^{2}} (and similarly for y {\displaystyle y} ); thus

10440-479: Is the identity matrix then ⟨ x , y ⟩ = x T M y {\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y} is the dot product. For another example, if n = 2 {\displaystyle n=2} and M = [ a b b d ] {\displaystyle \mathbf {M} ={\begin{bmatrix}a&b\\b&d\end{bmatrix}}}

10585-489: Is the transpose of x . {\displaystyle x.} A function ⟨ ⋅ , ⋅ ⟩ : R n × R n → R {\displaystyle \langle \,\cdot ,\cdot \,\rangle :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} } is an inner product on R n {\displaystyle \mathbb {R} ^{n}} if and only if there exists

10730-432: Is the dual basis element to e i {\displaystyle e_{i}} . The inverse is a component map These definitions make it manifest that there is a bijection The space of isomorphisms admits actions of orthogonal groups at either the V {\displaystyle V} side or the R n {\displaystyle \mathbb {R} ^{n}} side. For concreteness we fix

10875-419: Is the required system of orthogonal vectors, and the normalized vectors e 1 , … , e k {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}} form an orthonormal set . The calculation of the sequence u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}

11020-419: Is the same as that of u 1 , … , u n {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}} . If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the i {\displaystyle i} th step, assuming that v i {\displaystyle \mathbf {v} _{i}}

11165-480: Is the same as the subspace generated by v 1 , … , v i − 1 {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}} . The vector u i {\displaystyle \mathbf {u} _{i}} is then defined to be the difference between v i {\displaystyle \mathbf {v} _{i}} and this projection, guaranteed to be orthogonal to all of

11310-418: Is the zero vector in G . {\displaystyle G.} Hence the dimension of G {\displaystyle G} is | E | = ℵ 0 , {\displaystyle |E|=\aleph _{0},} whereas it is clear that the dimension of V {\displaystyle V} is c . {\displaystyle c.} This completes

11455-468: Is thus a one-to-one correspondence between complex inner products on a complex vector space V , {\displaystyle V,} and real inner products on V . {\displaystyle V.} For example, suppose that V = C n {\displaystyle V=\mathbb {C} ^{n}} for some integer n > 0. {\displaystyle n>0.} When V {\displaystyle V}

11600-851: Is used in the previous animation, when the intermediate v 3 ′ {\displaystyle \mathbf {v} '_{3}} vector is used when orthogonalizing the blue vector v 3 {\displaystyle \mathbf {v} _{3}} . Here is another description of the modified algorithm. Given the vectors v 1 , v 2 , … , v n {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}} , in our first step we produce vectors v 1 , v 2 ( 1 ) , … , v n ( 1 ) {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}} by removing components along

11745-513: The symmetric map ⟨ x , y ⟩ = x y {\displaystyle \langle x,y\rangle =xy} (rather than the usual conjugate symmetric map ⟨ x , y ⟩ = x y ¯ {\displaystyle \langle x,y\rangle =x{\overline {y}}} ) then its real part ⟨ x , y ⟩ R {\displaystyle \langle x,y\rangle _{\mathbb {R} }} would not be

11890-690: The Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} is orthonormal if ⟨ e i , e j ⟩ = 0 {\displaystyle \langle e_{i},e_{j}\rangle =0} for every i ≠ j {\displaystyle i\neq j} and ⟨ e i , e i ⟩ = ‖ e

12035-452: The Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that Theorem. Any complete inner product space has an orthonormal basis. The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is

12180-430: The completion of the span of { u β : β < min ( α , κ ) } {\displaystyle \{u_{\beta }:\beta <\min(\alpha ,\kappa )\}} is the same as that of { v β : β < α } {\displaystyle \{v_{\beta }:\beta <\alpha \}} . In particular, when applied to

12325-857: The dot product is an inner product space, an example of a Euclidean vector space . ⟨ [ x 1 ⋮ x n ] , [ y 1 ⋮ y n ] ⟩ = x T y = ∑ i = 1 n x i y i = x 1 y 1 + ⋯ + x n y n , {\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle =x^{\textsf {T}}y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},} where x T {\displaystyle x^{\operatorname {T} }}

12470-1007: The dot product of two vectors is 0 then they are orthogonal. For non-zero vectors, we can then normalize the vectors by dividing out their sizes as shown above: e 1 = 1 10 [ 3 1 ] {\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3\\1\end{bmatrix}}} e 2 = 1 40 25 [ − 2 / 5 6 / 5 ] = 1 10 [ − 1 3 ] . {\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {40 \over 25}}}{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1\\3\end{bmatrix}}.} Denote by GS ⁡ ( v 1 , … , v k ) {\displaystyle \operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})}

12615-721: The imaginary part (also called the complex part ) of ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is always 0. {\displaystyle 0.} Assume for the rest of this section that V {\displaystyle V} is a complex vector space. The polarization identity for complex vector spaces shows that The map defined by ⟨ x ∣ y ⟩ = ⟨ y , x ⟩ {\displaystyle \langle x\mid y\rangle =\langle y,x\rangle } for all x , y ∈ V {\displaystyle x,y\in V} satisfies

12760-424: The inner product of the vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } . This means that proj u ⁡ ( v ) {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} is the orthogonal projection of v {\displaystyle \mathbf {v} } onto

12905-540: The probability of the event. This definition of expectation as inner product can be extended to random vectors as well. The inner product for complex square matrices of the same size is the Frobenius inner product ⟨ A , B ⟩ := tr ⁡ ( A B † ) {\displaystyle \langle A,B\rangle :=\operatorname {tr} \left(AB^{\dagger }\right)} . Since trace and transposition are linear and

13050-1907: The row operation of adding a scalar multiple of one row to another. For example, taking v 1 = [ 3 1 ] , v 2 = [ 2 2 ] {\displaystyle \mathbf {v} _{1}={\begin{bmatrix}3&1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2&2\end{bmatrix}}} as above, we have [ A A T | A ] = [ 10 8 3 1 8 8 2 2 ] {\displaystyle \left[AA^{\mathsf {T}}|A\right]=\left[{\begin{array}{rr|rr}10&8&3&1\\8&8&2&2\end{array}}\right]} And reducing this to row echelon form produces [ 1 .8 .3 .1 0 1 − .25 .75 ] {\displaystyle \left[{\begin{array}{rr|rr}1&.8&.3&.1\\0&1&-.25&.75\end{array}}\right]} The normalized vectors are then e 1 = 1 .3 2 + .1 2 [ .3 .1 ] = 1 10 [ 3 1 ] {\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {.3^{2}+.1^{2}}}}{\begin{bmatrix}.3&.1\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3&1\end{bmatrix}}} e 2 = 1 .25 2 + .75 2 [ − .25 .75 ] = 1 10 [ − 1 3 ] , {\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {.25^{2}+.75^{2}}}}{\begin{bmatrix}-.25&.75\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1&3\end{bmatrix}},} as in

13195-422: The topology defined by the norm. In this article, F denotes a field that is either the real numbers R , {\displaystyle \mathbb {R} ,} or the complex numbers C . {\displaystyle \mathbb {C} .} A scalar is thus an element of F . A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector

13340-493: The Frobenius inner product is positive definite too, and so is an inner product. On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism V → V ∗ {\displaystyle V\to V^{*}} ), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of

13485-3290: The Gram–Schmidt process defines the vectors u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} as follows: u 1 = v 1 , e 1 = u 1 ‖ u 1 ‖ u 2 = v 2 − proj u 1 ⁡ ( v 2 ) , e 2 = u 2 ‖ u 2 ‖ u 3 = v 3 − proj u 1 ⁡ ( v 3 ) − proj u 2 ⁡ ( v 3 ) , e 3 = u 3 ‖ u 3 ‖ u 4 = v 4 − proj u 1 ⁡ ( v 4 ) − proj u 2 ⁡ ( v 4 ) − proj u 3 ⁡ ( v 4 ) , e 4 = u 4 ‖ u 4 ‖     ⋮     ⋮ u k = v k − ∑ j = 1 k − 1 proj u j ⁡ ( v k ) , e k = u k ‖ u k ‖ . {\displaystyle {\begin{aligned}\mathbf {u} _{1}&=\mathbf {v} _{1},&\!\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\|\mathbf {u} _{1}\|}}\\\mathbf {u} _{2}&=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2}),&\!\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\|\mathbf {u} _{2}\|}}\\\mathbf {u} _{3}&=\mathbf {v} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{3})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{3}),&\!\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\|\mathbf {u} _{3}\|}}\\\mathbf {u} _{4}&=\mathbf {v} _{4}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{3}}(\mathbf {v} _{4}),&\!\mathbf {e} _{4}&={\mathbf {u} _{4} \over \|\mathbf {u} _{4}\|}\\&{}\ \ \vdots &&{}\ \ \vdots \\\mathbf {u} _{k}&=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}(\mathbf {v} _{k}),&\!\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\|\mathbf {u} _{k}\|}}.\end{aligned}}} The sequence u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}

13630-484: The above formula for u 2 {\displaystyle \mathbf {u} _{2}} : we get zero. Then use this to compute ⟨ u 1 , u 3 ⟩ {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{3}\rangle } again by substituting the formula for u 3 {\displaystyle \mathbf {u} _{3}} : we get zero. For arbitrary k {\displaystyle k}

13775-506: The action again given by composition: C ∗ R i j = C ∘ R i j {\displaystyle C*R_{ij}=C\circ R_{ij}} . The set of orthonormal bases for R n {\displaystyle \mathbb {R} ^{n}} with the standard inner product is a principal homogeneous space or G-torsor for the orthogonal group G = O ( n ) , {\displaystyle G={\text{O}}(n),} and

13920-548: The action given by composition: R ∗ C = R ∘ C . {\displaystyle R*C=R\circ C.} This space also admits a right action by the group of isometries of R n {\displaystyle \mathbb {R} ^{n}} , that is, R i j ∈ O ( n ) ⊂ Mat n × n ( R ) {\displaystyle R_{ij}\in {\text{O}}(n)\subset {\text{Mat}}_{n\times n}(\mathbb {R} )} , with

14065-783: The algorithm will then be the dimension of the space spanned by the original inputs. A variant of the Gram–Schmidt process using transfinite recursion applied to a (possibly uncountably) infinite sequence of vectors ( v α ) α < λ {\displaystyle (v_{\alpha })_{\alpha <\lambda }} yields a set of orthonormal vectors ( u α ) α < κ {\displaystyle (u_{\alpha })_{\alpha <\kappa }} with κ ≤ λ {\displaystyle \kappa \leq \lambda } such that for any α ≤ λ {\displaystyle \alpha \leq \lambda } ,

14210-626: The assignment x ↦ ⟨ x , x ⟩ {\displaystyle x\mapsto {\sqrt {\langle x,x\rangle }}} would not define a norm. The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if ⟨ x , y ⟩ = 0 {\displaystyle \langle x,y\rangle =0} then ⟨ x , y ⟩ R = 0 , {\displaystyle \langle x,y\rangle _{\mathbb {R} }=0,} but

14355-401: The augmented matrix [ A A T | A ] {\displaystyle \left[AA^{\mathsf {T}}|A\right]} will produce the orthogonalized vectors in place of A {\displaystyle A} . However the matrix A A T {\displaystyle AA^{\mathsf {T}}} must be brought to row echelon form , using only

14500-505: The axioms of the inner product except that it is antilinear in its first , rather than its second, argument. The real part of both ⟨ x ∣ y ⟩ {\displaystyle \langle x\mid y\rangle } and ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle } are equal to Re ⁡ ⟨ x , y ⟩ {\displaystyle \operatorname {Re} \langle x,y\rangle } but

14645-614: The basis as a map ψ B : V → R n {\displaystyle \psi _{\mathcal {B}}:V\rightarrow \mathbb {R} ^{n}} which is an isomorphism of inner product spaces: to make this more explicit we can write Explicitly we can write ( ψ B ( v ) ) i = e i ( v ) = ϕ ( e i , v ) {\displaystyle (\psi _{\mathcal {B}}(v))^{i}=e^{i}(v)=\phi (e_{i},v)} where e i {\displaystyle e^{i}}

14790-631: The cardinality of the continuum, it must be that | F | = c . {\displaystyle |F|=c.} Let L {\displaystyle L} be a Hilbert space of dimension c {\displaystyle c} (for instance, L = ℓ 2 ( R ) {\displaystyle L=\ell ^{2}(\mathbb {R} )} ). Let B {\displaystyle B} be an orthonormal basis for L {\displaystyle L} and let φ : F → B {\displaystyle \varphi :F\to B} be

14935-564: The complex inner product ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle } is the map ⟨ x , y ⟩ R = Re ⁡ ⟨ x , y ⟩   :   V R × V R → R , {\displaystyle \langle x,y\rangle _{\mathbb {R} }=\operatorname {Re} \langle x,y\rangle ~:~V_{\mathbb {R} }\times V_{\mathbb {R} }\to \mathbb {R} ,} which necessarily forms

15080-605: The complex inner product gives ⟨ x , A x ⟩ = − i ‖ x ‖ 2 , {\displaystyle \langle x,Ax\rangle =-i\|x\|^{2},} which (as expected) is not identically zero. Let V {\displaystyle V} be a finite dimensional inner product space of dimension n . {\displaystyle n.} Recall that every basis of V {\displaystyle V} consists of exactly n {\displaystyle n} linearly independent vectors. Using

15225-457: The components of ϕ {\displaystyle \phi } are particularly simple: ϕ ( e i , e j ) = δ i j {\displaystyle \phi (e_{i},e_{j})=\delta _{ij}} (where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta ). We can now view

15370-929: The conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by, ⟨ A , B ⟩ = tr ⁡ ( A B † ) = tr ⁡ ( B A † ) ¯ = ⟨ B , A ⟩ ¯ {\displaystyle \langle A,B\rangle =\operatorname {tr} \left(AB^{\dagger }\right)={\overline {\operatorname {tr} \left(BA^{\dagger }\right)}}={\overline {\left\langle B,A\right\rangle }}} Finally, since for A {\displaystyle A} nonzero, ⟨ A , A ⟩ = ∑ i j | A i j | 2 > 0 {\displaystyle \langle A,A\rangle =\sum _{ij}\left|A_{ij}\right|^{2}>0} , we get that

15515-2415: The direction of v 1 {\displaystyle \mathbf {v} _{1}} . In formulas, v k ( 1 ) := v k − ⟨ v k , v 1 ⟩ ⟨ v 1 , v 1 ⟩ v 1 {\displaystyle \mathbf {v} _{k}^{(1)}:=\mathbf {v} _{k}-{\frac {\langle \mathbf {v} _{k},\mathbf {v} _{1}\rangle }{\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle }}\mathbf {v} _{1}} . After this step we already have two of our desired orthogonal vectors u 1 , … , u n {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}} , namely u 1 = v 1 , u 2 = v 2 ( 1 ) {\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1},\mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}} , but we also made v 3 ( 1 ) , … , v n ( 1 ) {\displaystyle \mathbf {v} _{3}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}} already orthogonal to u 1 {\displaystyle \mathbf {u} _{1}} . Next, we orthogonalize those remaining vectors against u 2 = v 2 ( 1 ) {\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}} . This means we compute v 3 ( 2 ) , v 4 ( 2 ) , … , v n ( 2 ) {\displaystyle \mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} by subtraction v k ( 2 ) := v k ( 1 ) − ⟨ v k ( 1 ) , u 2 ⟩ ⟨ u 2 , u 2 ⟩ u 2 {\displaystyle \mathbf {v} _{k}^{(2)}:=\mathbf {v} _{k}^{(1)}-{\frac {\langle \mathbf {v} _{k}^{(1)},\mathbf {u} _{2}\rangle }{\langle \mathbf {u} _{2},\mathbf {u} _{2}\rangle }}\mathbf {u} _{2}} . Now we have stored

15660-459: The dot product; furthermore, without the complex conjugate, if x ∈ C {\displaystyle x\in \mathbb {C} } but x ∉ R {\displaystyle x\not \in \mathbb {R} } then ⟨ x , x ⟩ = x x = x 2 ∉ [ 0 , ∞ ) {\displaystyle \langle x,x\rangle =xx=x^{2}\not \in [0,\infty )} so

15805-4058: The example above. The result of the Gram–Schmidt process may be expressed in a non-recursive formula using determinants . e j = 1 D j − 1 D j | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v 2 , v 2 ⟩ ⋯ ⟨ v j , v 2 ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v 1 , v j − 1 ⟩ ⟨ v 2 , v j − 1 ⟩ ⋯ ⟨ v j , v j − 1 ⟩ v 1 v 2 ⋯ v j | {\displaystyle \mathbf {e} _{j}={\frac {1}{\sqrt {D_{j-1}D_{j}}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}} u j = 1 D j − 1 | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v 2 , v 2 ⟩ ⋯ ⟨ v j , v 2 ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v 1 , v j − 1 ⟩ ⟨ v 2 , v j − 1 ⟩ ⋯ ⟨ v j , v j − 1 ⟩ v 1 v 2 ⋯ v j | {\displaystyle \mathbf {u} _{j}={\frac {1}{D_{j-1}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}} where D 0 = 1 {\displaystyle D_{0}=1} and, for j ≥ 1 {\displaystyle j\geq 1} , D j {\displaystyle D_{j}}

15950-424: The expression for u k {\displaystyle \mathbf {u} _{k}} is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a cofactor expansion along the row of vectors. The determinant formula for the Gram-Schmidt is computationally (exponentially) slower than the recursive algorithms described above; it

16095-955: The following properties: Let g : R n → R n {\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} be orthogonal (with respect to the given inner product). Then we have GS ⁡ ( g ( v 1 ) , … , g ( v k ) ) = ( g ( GS ⁡ ( v 1 , … , v k ) 1 ) , … , g ( GS ⁡ ( v 1 , … , v k ) k ) ) {\displaystyle \operatorname {GS} (g(\mathbf {v} _{1}),\dots ,g(\mathbf {v} _{k}))=\left(g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{1}),\dots ,g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{k})\right)} Further,

16240-767: The following properties: Suppose that ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is an inner product on V {\displaystyle V} (so it is antilinear in its second argument). The polarization identity shows that the real part of the inner product is Re ⁡ ⟨ x , y ⟩ = 1 4 ( ‖ x + y ‖ 2 − ‖ x − y ‖ 2 ) . {\displaystyle \operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right).} If V {\displaystyle V}

16385-488: The full set of orthogonal vectors u 1 , … , u n {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}} . If orthonormal vectors are desired, then we normalize as we go, so that the denominators in the subtraction formulas turn into ones. The following MATLAB algorithm implements classical Gram–Schmidt orthonormalization. The vectors v 1 , ..., v k (columns of matrix V , so that V(:,j)

16530-1850: The graph of T . {\displaystyle T.} Let G ¯ {\displaystyle {\overline {G}}} be the closure of G {\displaystyle G} in V {\displaystyle V} ; we will show G ¯ = V . {\displaystyle {\overline {G}}=V.} Since for any e ∈ E {\displaystyle e\in E} we have ( e , 0 ) ∈ G , {\displaystyle (e,0)\in G,} it follows that K ⊕ 0 ⊆ G ¯ . {\displaystyle K\oplus 0\subseteq {\overline {G}}.} Next, if b ∈ B , {\displaystyle b\in B,} then b = T f {\displaystyle b=Tf} for some f ∈ F ⊆ K , {\displaystyle f\in F\subseteq K,} so ( f , b ) ∈ G ⊆ G ¯ {\displaystyle (f,b)\in G\subseteq {\overline {G}}} ; since ( f , 0 ) ∈ G ¯ {\displaystyle (f,0)\in {\overline {G}}} as well, we also have ( 0 , b ) ∈ G ¯ . {\displaystyle (0,b)\in {\overline {G}}.} It follows that 0 ⊕ L ⊆ G ¯ , {\displaystyle 0\oplus L\subseteq {\overline {G}},} so G ¯ = V , {\displaystyle {\overline {G}}=V,} and G {\displaystyle G}

16675-595: The inner product is the dot product or scalar product of Cartesian coordinates . Inner product spaces of infinite dimension are widely used in functional analysis . Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces . The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano , in 1898. An inner product naturally induces an associated norm , (denoted | x | {\displaystyle |x|} and | y | {\displaystyle |y|} in

16820-442: The inner product). Say that E {\displaystyle E} is an orthonormal basis for V {\displaystyle V} if it is a basis and ⟨ e a , e b ⟩ = 0 {\displaystyle \left\langle e_{a},e_{b}\right\rangle =0} if a ≠ b {\displaystyle a\neq b} and ⟨ e

16965-399: The inner products differ in their complex part: The last equality is similar to the formula expressing a linear functional in terms of its real part. These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on V , {\displaystyle V,} considered as a real vector space. There

17110-488: The interval [ − 1 , 1 ] {\displaystyle [-1,1]} can be expressed ( almost everywhere ) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials x n . {\displaystyle x^{n}.} A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces M {\displaystyle M} equipped with

17255-688: The interval [−1, 1] the sequence of continuous "step" functions, { f k } k , {\displaystyle \{f_{k}\}_{k},} defined by: f k ( t ) = { 0 t ∈ [ − 1 , 0 ] 1 t ∈ [ 1 k , 1 ] k t t ∈ ( 0 , 1 k ) {\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}} This sequence

17400-779: The isomorphisms to point in the direction R n → V {\displaystyle \mathbb {R} ^{n}\rightarrow V} , and consider the space of such maps, Iso ( R n → V ) {\displaystyle {\text{Iso}}(\mathbb {R} ^{n}\rightarrow V)} . This space admits a left action by the group of isometries of V {\displaystyle V} , that is, R ∈ GL ( V ) {\displaystyle R\in {\text{GL}}(V)} such that ϕ ( ⋅ , ⋅ ) = ϕ ( R ⋅ , R ⋅ ) {\displaystyle \phi (\cdot ,\cdot )=\phi (R\cdot ,R\cdot )} , with

17545-456: The larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice . However, one would have to use the axiom of countable choice .) For concreteness we discuss orthonormal bases for a real, n {\displaystyle n} -dimensional vector space V {\displaystyle V} with

17690-550: The line spanned by u {\displaystyle \mathbf {u} } . If u {\displaystyle \mathbf {u} } is the zero vector, then proj u ⁡ ( v ) {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} is defined as the zero vector. Given k {\displaystyle k} vectors v 1 , … , v k {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}}

17835-410: The metric induced by the inner product yields a complete metric space . An example of an inner product space which induces an incomplete metric is the space C ( [ a , b ] ) {\displaystyle C([a,b])} of continuous complex valued functions f {\displaystyle f} and g {\displaystyle g} on the interval [

17980-623: The next example shows that the converse is in general not true. Given any x ∈ V , {\displaystyle x\in V,} the vector i x {\displaystyle ix} (which is the vector x {\displaystyle x} rotated by 90°) belongs to V {\displaystyle V} and so also belongs to V R {\displaystyle V_{\mathbb {R} }} (although scalar multiplication of x {\displaystyle x} by i = − 1 {\displaystyle i={\sqrt {-1}}}

18125-404: The orthogonal group, but not principal homogeneous spaces: any k {\displaystyle k} -frame can be taken to any other k {\displaystyle k} -frame by an orthogonal map, but this map is not uniquely determined. Inner product In mathematics , an inner product space (or, rarely, a Hausdorff pre-Hilbert space ) is a real vector space or

18270-442: The picture); so, every inner product space is a normed vector space . If this normed space is also complete (that is, a Banach space ) then the inner product space is a Hilbert space . If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space H ¯ . {\displaystyle {\overline {H}}.} This means that H {\displaystyle H}

18415-549: The proof is accomplished by mathematical induction . Geometrically, this method proceeds as follows: to compute u i {\displaystyle \mathbf {u} _{i}} , it projects v i {\displaystyle \mathbf {v} _{i}} orthogonally onto the subspace U {\displaystyle U} generated by u 1 , … , u i − 1 {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{i-1}} , which

18560-614: The proof. Parseval's identity leads immediately to the following theorem: Theorem. Let V {\displaystyle V} be a separable inner product space and { e k } k {\displaystyle \left\{e_{k}\right\}_{k}} an orthonormal basis of V . {\displaystyle V.} Then the map x ↦ { ⟨ e k , x ⟩ } k ∈ N {\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}

18705-437: The real inner product the value is always ⟨ x , i x ⟩ R = 0. {\displaystyle \langle x,ix\rangle _{\mathbb {R} }=0.} If ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is a complex inner product and A : V → V {\displaystyle A:V\to V}

18850-413: The real numbers, the assignment ( x , y ) ↦ x y {\displaystyle (x,y)\mapsto xy} does not define a complex inner product on C . {\displaystyle \mathbb {C} .} More generally, the real n {\displaystyle n} -space R n {\displaystyle \mathbb {R} ^{n}} with

18995-462: The real part of this map ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is equal to the dot product). Real vs. complex inner products Let V R {\displaystyle V_{\mathbb {R} }} denote V {\displaystyle V} considered as a vector space over the real numbers rather than complex numbers. The real part of

19140-610: The remaining vectors are already orthogonal to u 1 , u 2 {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2}} . As should be clear now, the next step orthogonalizes v 4 ( 2 ) , … , v n ( 2 ) {\displaystyle \mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} against u 3 = v 3 ( 2 ) {\displaystyle \mathbf {u} _{3}=\mathbf {v} _{3}^{(2)}} . Proceeding in this manner we find

19285-486: The result of applying the Gram–Schmidt process to a collection of vectors v 1 , … , v k {\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}} . This yields a map GS : ( R n ) k → ( R n ) k {\displaystyle \operatorname {GS} \colon (\mathbb {R} ^{n})^{k}\to (\mathbb {R} ^{n})^{k}} . It has

19430-405: The same k {\displaystyle k} -dimensional subspace of R n {\displaystyle \mathbb {R} ^{n}} as S {\displaystyle S} . The method is named after Jørgen Pedersen Gram and Erhard Schmidt , but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions , it

19575-618: The same subspace as the columns of the original matrix V {\displaystyle V} . The explicit use of the product V ∗ V {\displaystyle V^{*}V} makes the algorithm unstable, especially if the product's condition number is large. Nevertheless, this algorithm is used in practice and implemented in some software packages because of its high efficiency and simplicity. In quantum mechanics there are several orthogonalization schemes with characteristics better suited for certain applications than original Gram–Schmidt. Nevertheless, it remains

19720-1066: The second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. Bra-ket notation in quantum mechanics also uses slightly different notation, i.e. ⟨ ⋅ | ⋅ ⟩ {\displaystyle \langle \cdot |\cdot \rangle } , where ⟨ x | y ⟩ := ( y , x ) {\displaystyle \langle x|y\rangle :=\left(y,x\right)} . Several notations are used for inner products, including ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , ( ⋅ , ⋅ ) {\displaystyle \left(\cdot ,\cdot \right)} , ⟨ ⋅ | ⋅ ⟩ {\displaystyle \langle \cdot |\cdot \rangle } and ( ⋅ | ⋅ ) {\displaystyle \left(\cdot |\cdot \right)} , as well as

19865-503: The smallest closed linear subspace V ⊆ H {\displaystyle V\subseteq H} containing S . {\displaystyle S.} Then S {\displaystyle S} will be an orthonormal basis of V ; {\displaystyle V;} which may of course be smaller than H {\displaystyle H} itself, being an incomplete orthonormal set, or be H , {\displaystyle H,} when it

20010-448: The standard basis under a rotation or reflection (or any orthogonal transformation ) is also orthonormal, and every orthonormal basis for R n {\displaystyle \mathbb {R} ^{n}} arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization . The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame . For

20155-422: The standard inner product ⟨ x , y ⟩ = x y ¯ , {\displaystyle \langle x,y\rangle =x{\overline {y}},} on C {\displaystyle \mathbb {C} } is an "extension" the dot product . Also, had ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle } been instead defined to be

20300-414: The study of R n {\displaystyle \mathbb {R} ^{n}} under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process . In functional analysis , the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces . Given

20445-782: The usual dot product. Among the simplest examples of inner product spaces are R {\displaystyle \mathbb {R} } and C . {\displaystyle \mathbb {C} .} The real numbers R {\displaystyle \mathbb {R} } are a vector space over R {\displaystyle \mathbb {R} } that becomes an inner product space with arithmetic multiplication as its inner product: ⟨ x , y ⟩ := x y  for  x , y ∈ R . {\displaystyle \langle x,y\rangle :=xy\quad {\text{ for }}x,y\in \mathbb {R} .} The complex numbers C {\displaystyle \mathbb {C} } are

20590-761: The vectors u 1 {\displaystyle \mathbf {u} _{1}} and u 2 {\displaystyle \mathbf {u} _{2}} are indeed orthogonal: ⟨ u 1 , u 2 ⟩ = ⟨ [ 3 1 ] , [ − 2 / 5 6 / 5 ] ⟩ = − 6 5 + 6 5 = 0 , {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{bmatrix}3\\1\end{bmatrix}},{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0,} noting that if

20735-602: The vectors v 1 , v 2 ( 1 ) , v 3 ( 2 ) , v 4 ( 2 ) , … , v n ( 2 ) {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} where the first three vectors are already u 1 , u 2 , u 3 {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\mathbf {u} _{3}} and

20880-465: The vectors in the subspace U {\displaystyle U} . The Gram–Schmidt process also applies to a linearly independent countably infinite sequence { v i } i . The result is an orthogonal (or orthonormal) sequence { u i } i such that for natural number n : the algebraic span of v 1 , … , v n {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n}}

21025-540: The vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration . Yet another alternative is motivated by the use of Cholesky decomposition for inverting the matrix of the normal equations in linear least squares . Let V {\displaystyle V} be a full column rank matrix, whose columns need to be orthogonalized. The matrix V ∗ V {\displaystyle V^{*}V}

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