Linear algebra is the branch of mathematics concerning linear equations such as:
88-475: In linear algebra , an eigenvector ( / ˈ aɪ ɡ ən -/ EYE -gən- ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation . More precisely, an eigenvector, v {\displaystyle \mathbf {v} } , of a linear transformation, T {\displaystyle T} , is scaled by a constant factor , λ {\displaystyle \lambda } , when
176-407: A 2 , ..., a k are in F form a linear subspace called the span of S . The span of S is also the intersection of all linear subspaces containing S . In other words, it is the smallest (for the inclusion relation) linear subspace containing S . A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if
264-455: A 1 , a 2 , ..., a n ) where a i ∈ K and n is the dimension of the vector space in consideration.). For example, every real vector space of dimension n is isomorphic to the n -dimensional real space R . Alternatively, a vector space V can be equipped with a norm function that assigns to every vector v in V a scalar || v ||. By definition, multiplying v by a scalar k also multiplies its norm by | k |. If || v ||
352-489: A (linear) function space , kf is the function x ↦ k ( f ( x )) . The scalars can be taken from any field, including the rational , algebraic , real, and complex numbers, as well as finite fields . According to a fundamental theorem of linear algebra, every vector space has a basis . It follows that every vector space over a field K is isomorphic to the corresponding coordinate vector space where each coordinate consists of elements of K (E.g., coordinates (
440-566: A (unitary) matrix V {\displaystyle V} whose first γ A ( λ ) {\displaystyle \gamma _{A}(\lambda )} columns are these eigenvectors, and whose remaining columns can be any orthonormal set of n − γ A ( λ ) {\displaystyle n-\gamma _{A}(\lambda )} vectors orthogonal to these eigenvectors of A {\displaystyle A} . Then V {\displaystyle V} has full rank and
528-447: A difference w – z , and the line segments wz and 0( w − z ) are of the same length and direction. The segments are equipollent . The four-dimensional system H {\displaystyle \mathbb {H} } of quaternions was discovered by W.R. Hamilton in 1843. The term vector was introduced as v = x i + y j + z k representing a point in space. The quaternion difference p – q also produces
616-777: A distinct eigenvalue and raised to the power of the algebraic multiplicity, det ( A − λ I ) = ( λ 1 − λ ) μ A ( λ 1 ) ( λ 2 − λ ) μ A ( λ 2 ) ⋯ ( λ d − λ ) μ A ( λ d ) . {\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )^{\mu _{A}(\lambda _{1})}(\lambda _{2}-\lambda )^{\mu _{A}(\lambda _{2})}\cdots (\lambda _{d}-\lambda )^{\mu _{A}(\lambda _{d})}.} If d = n then
704-652: A factor ( ξ − λ ) γ A ( λ ) {\displaystyle (\xi -\lambda )^{\gamma _{A}(\lambda )}} , which means that the algebraic multiplicity of λ {\displaystyle \lambda } must satisfy μ A ( λ ) ≥ γ A ( λ ) {\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )} . Linear algebra linear maps such as: and their representations in vector spaces and through matrices . Linear algebra
792-508: A factor of λ , where λ is a scalar , then v {\displaystyle \mathbf {v} } is called an eigenvector of A , and λ is the corresponding eigenvalue. This relationship can be expressed as: A v = λ v {\displaystyle A\mathbf {v} =\lambda \mathbf {v} } . There is a direct correspondence between n -by- n square matrices and linear transformations from an n -dimensional vector space into itself, given any basis of
880-503: A few years later. At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen , which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Hermann von Helmholtz . For some time, the standard term in English
968-473: A finite number of elements, V is a finite-dimensional vector space . If U is a subspace of V , then dim U ≤ dim V . In the case where V is finite-dimensional, the equality of the dimensions implies U = V . If U 1 and U 2 are subspaces of V , then where U 1 + U 2 denotes the span of U 1 ∪ U 2 . Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps . Their theory
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#17328525060531056-398: A linear map T : V → W , the image T ( V ) of V , and the inverse image T ( 0 ) of 0 (called kernel or null space), are linear subspaces of W and V , respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums where v 1 , v 2 , ..., v k are in S , and a 1 ,
1144-407: A linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ , called an eigenvalue. This condition can be written as the equation T ( v ) = λ v , {\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,} referred to as
1232-503: A matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants". Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later. The telegraph required an explanatory system, and
1320-404: A part of the basis of W is mapped bijectively on a part of the basis of V , and that the remaining basis elements of W , if any, are mapped to zero. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results. A finite set of linear equations in a finite set of variables, for example, x 1 , x 2 , ..., x n , or x , y , ..., z
1408-408: A segment equipollent to pq . Other hypercomplex number systems also used the idea of a linear space with a basis . Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group . The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote
1496-449: A system, one may associate its matrix and its right member vector Let T be the linear transformation associated to the matrix M . A solution of the system ( S ) is a vector Scalar (mathematics) A scalar is an element of a field which is used to define a vector space . In linear algebra , real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through
1584-405: A vector by its inverse image under this isomorphism, that is by the coordinate vector ( a 1 , ..., a m ) or by the column matrix If W is another finite dimensional vector space (possibly the same), with a basis ( w 1 , ..., w n ) , a linear map f from W to V is well defined by its values on the basis elements, that is ( f ( w 1 ), ..., f ( w n )) . Thus, f
1672-530: A vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra . The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. Until
1760-418: A vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space . A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector . The term scalar is also sometimes used informally to mean a vector, matrix , tensor , or other, usually, "compound" value that
1848-457: Is a set V equipped with two binary operations . Elements of V are called vectors , and elements of F are called scalars . The first operation, vector addition , takes any two vectors v and w and outputs a third vector v + w . The second operation, scalar multiplication , takes any scalar a and any vector v and outputs a new vector a v . The axioms that addition and scalar multiplication must satisfy are
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#17328525060531936-513: Is a complex number and the eigenvectors are complex n by 1 matrices. A property of the nullspace is that it is a linear subspace , so E is a linear subspace of C n {\displaystyle \mathbb {C} ^{n}} . Because the eigenspace E is a linear subspace, it is closed under addition. That is, if two vectors u and v belong to the set E , written u , v ∈ E , then ( u + v ) ∈ E or equivalently A ( u + v ) = λ ( u + v ) . This can be checked using
2024-454: Is a complex number. The numbers λ 1 , λ 2 , ..., λ n , which may not all have distinct values, are roots of the polynomial and are the eigenvalues of A . As a brief example, which is described in more detail in the examples section later, consider the matrix A = [ 2 1 1 2 ] . {\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.} Taking
2112-1932: Is a scalar λ such that x = λ y . {\displaystyle \mathbf {x} =\lambda \mathbf {y} .} In this case, λ = − 1 20 {\displaystyle \lambda =-{\frac {1}{20}}} . Now consider the linear transformation of n -dimensional vectors defined by an n by n matrix A , A v = w , {\displaystyle A\mathbf {v} =\mathbf {w} ,} or [ A 11 A 12 ⋯ A 1 n A 21 A 22 ⋯ A 2 n ⋮ ⋮ ⋱ ⋮ A n 1 A n 2 ⋯ A n n ] [ v 1 v 2 ⋮ v n ] = [ w 1 w 2 ⋮ w n ] {\displaystyle {\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1n}\\A_{21}&A_{22}&\cdots &A_{2n}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nn}\\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\\\vdots \\w_{n}\end{bmatrix}}} where, for each row, w i = A i 1 v 1 + A i 2 v 2 + ⋯ + A i n v n = ∑ j = 1 n A i j v j . {\displaystyle w_{i}=A_{i1}v_{1}+A_{i2}v_{2}+\cdots +A_{in}v_{n}=\sum _{j=1}^{n}A_{ij}v_{j}.} If it occurs that v and w are scalar multiples, that
2200-428: Is a spanning set such that S ⊆ T , then there is a basis B such that S ⊆ B ⊆ T . Any two bases of a vector space V have the same cardinality , which is called the dimension of V ; this is the dimension theorem for vector spaces . Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. If any basis of V (and therefore every basis) has
2288-449: Is actually reduced to a single component. Thus, for example, the product of a 1 × n matrix and an n × 1 matrix, which is formally a 1 × 1 matrix, is often said to be a scalar . The real component of a quaternion is also called its scalar part . The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix . The word scalar derives from
2376-491: Is adopted from the German word eigen ( cognate with the English word own ) for 'proper', 'characteristic', 'own'. Originally used to study principal axes of the rotational motion of rigid bodies , eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis , vibration analysis , atomic orbitals , facial recognition , and matrix diagonalization . In essence, an eigenvector v of
2464-429: Is always (−1) λ . This polynomial is called the characteristic polynomial of A . Equation ( 3 ) is called the characteristic equation or the secular equation of A . The fundamental theorem of algebra implies that the characteristic polynomial of an n -by- n matrix A , being a polynomial of degree n , can be factored into the product of n linear terms, where each λ i may be real but in general
2552-424: Is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either. Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so
2640-454: Is called a shear mapping . Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation
2728-425: Is called a system of linear equations or a linear system . Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. For example, let be a linear system. To such
Eigenvalues and eigenvectors - Misplaced Pages Continue
2816-594: Is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry , including for defining basic objects such as lines , planes and rotations . Also, functional analysis , a branch of mathematical analysis , may be viewed as the application of linear algebra to function spaces . Linear algebra is also used in most sciences and fields of engineering , because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems , which cannot be modeled with linear algebra, it
2904-404: Is if then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector. Equation ( 1 ) is the eigenvalue equation for the matrix A . Equation ( 1 ) can be stated equivalently as where I is the n by n identity matrix and 0 is the zero vector. Equation ( 2 ) has a nonzero solution v if and only if
2992-443: Is interpreted as the length of v , this operation can be described as scaling the length of v by k . A vector space equipped with a norm is called a normed vector space (or normed linear space ). The norm is usually defined to be an element of V 's scalar field K , which restricts the latter to fields that support the notion of sign. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as
3080-437: Is often used for dealing with first-order approximations , using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point. The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on
3168-451: Is precisely the kernel or nullspace of the matrix ( A − λI ). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ . So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ , and E equals the nullspace of ( A − λI ). E is called the eigenspace or characteristic space of A associated with λ . In general λ
3256-628: Is said to be a simple eigenvalue . If μ A ( λ i ) equals the geometric multiplicity of λ i , γ A ( λ i ), defined in the next section, then λ i is said to be a semisimple eigenvalue . Given a particular eigenvalue λ of the n by n matrix A , define the set E to be all vectors v that satisfy equation ( 2 ), E = { v : ( A − λ I ) v = 0 } . {\displaystyle E=\left\{\mathbf {v} :\left(A-\lambda I\right)\mathbf {v} =\mathbf {0} \right\}.} On one hand, this set
3344-482: Is similar to D − ξ I {\displaystyle D-\xi I} , and det ( A − ξ I ) = det ( D − ξ I ) {\displaystyle \det(A-\xi I)=\det(D-\xi I)} . But from the definition of D {\displaystyle D} , we know that det ( D − ξ I ) {\displaystyle \det(D-\xi I)} contains
3432-927: Is therefore invertible. Evaluating D := V T A V {\displaystyle D:=V^{T}AV} , we get a matrix whose top left block is the diagonal matrix λ I γ A ( λ ) {\displaystyle \lambda I_{\gamma _{A}(\lambda )}} . This can be seen by evaluating what the left-hand side does to the first column basis vectors. By reorganizing and adding − ξ V {\displaystyle -\xi V} on both sides, we get ( A − ξ I ) V = V ( D − ξ I ) {\displaystyle (A-\xi I)V=V(D-\xi I)} since I {\displaystyle I} commutes with V {\displaystyle V} . In other words, A − ξ I {\displaystyle A-\xi I}
3520-568: Is thus an essential part of linear algebra. Let V be a finite-dimensional vector space over a field F , and ( v 1 , v 2 , ..., v m ) be a basis of V (thus m is the dimension of V ). By definition of a basis, the map is a bijection from F , the set of the sequences of m elements of F , onto V . This is an isomorphism of vector spaces, if F is equipped of its standard structure of vector space, where vector addition and scalar multiplication are done component by component. This isomorphism allows representing
3608-415: Is well represented by the list of the corresponding column matrices. That is, if for j = 1, ..., n , then f is represented by the matrix with m rows and n columns. Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing
Eigenvalues and eigenvectors - Misplaced Pages Continue
3696-613: The Latin word scalaris , an adjectival form of scala (Latin for "ladder"), from which the English word scale also comes. The first recorded usage of the word "scalar" in mathematics occurs in François Viète 's Analytic Art ( In artem analyticem isagoge ) (1591): According to a citation in the Oxford English Dictionary the first recorded usage of the term "scalar" in English came with W. R. Hamilton in 1846, referring to
3784-472: The determinant of the matrix ( A − λI ) is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation Using the Leibniz formula for determinants , the left-hand side of equation ( 3 ) is a polynomial function of the variable λ and the degree of this polynomial is n , the order of the matrix A . Its coefficients depend on the entries of A , except that its term of degree n
3872-482: The distributive property of matrix multiplication. Similarly, because E is a linear subspace, it is closed under scalar multiplication. That is, if v ∈ E and α is a complex number, ( α v ) ∈ E or equivalently A ( α v ) = λ ( α v ) . This can be checked by noting that multiplication of complex matrices by complex numbers is commutative . As long as u + v and α v are not zero, they are also eigenvectors of A associated with λ . The dimension of
3960-506: The eigenvalue equation or eigenequation . In general, λ may be any scalar . For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex . The example here, based on the Mona Lisa , provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example
4048-540: The quadric surfaces , and generalized it to arbitrary dimensions. Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue ; his term survives in characteristic equation . Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his 1822 treatise The Analytic Theory of Heat (Théorie analytique de la chaleur) . Charles-François Sturm elaborated on Fourier's ideas further, and brought them to
4136-446: The stability theory started by Laplace, by realizing that defective matrices can cause instability. In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory . Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation
4224-522: The 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression. Linear algebra is flat differential geometry and serves in tangent spaces to manifolds . Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations , and much of the history of linear algebra is the history of Lorentz transformations . The first modern and more precise definition of
4312-406: The 19th century, linear algebra was introduced through systems of linear equations and matrices . In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic , more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. A vector space over a field F (often the field of the real numbers )
4400-807: The Mathematical Art . Its use is illustrated in eighteen problems, with two to five equations. Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry . In fact, in this new geometry, now called Cartesian geometry , lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule . Later, Gauss further described
4488-406: The areas where linear algebra is applied, from geology to quantum mechanics . In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation ( feedback ). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of
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#17328525060534576-499: The attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices . Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle , and Alfred Clebsch found the corresponding result for skew-symmetric matrices . Finally, Karl Weierstrass clarified an important aspect in
4664-416: The characteristic polynomial equal to zero, it has roots at λ=1 and λ=3 , which are the two eigenvalues of A . The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation ( A − λ I ) v = 0 {\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} } . In this example,
4752-845: The context of linear algebra courses focused on matrices. Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications. Consider n -dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors x = [ 1 − 3 4 ] and y = [ − 20 60 − 80 ] . {\displaystyle \mathbf {x} ={\begin{bmatrix}1\\-3\\4\end{bmatrix}}\quad {\mbox{and}}\quad \mathbf {y} ={\begin{bmatrix}-20\\60\\-80\end{bmatrix}}.} These vectors are said to be scalar multiples of each other, or parallel or collinear , if there
4840-484: The determinant of ( A − λI ) , the characteristic polynomial of A is det ( A − λ I ) = | 2 − λ 1 1 2 − λ | = 3 − 4 λ + λ 2 . {\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.} Setting
4928-408: The eigenspace E associated with λ , or equivalently the maximum number of linearly independent eigenvectors associated with λ , is referred to as the eigenvalue's geometric multiplicity γ A ( λ ) {\displaystyle \gamma _{A}(\lambda )} . Because E is also the nullspace of ( A − λI ), the geometric multiplicity of λ is the dimension of
5016-436: The eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix —for example by diagonalizing it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them: Eigenvalues are often introduced in the context of linear algebra or matrix theory . Historically, however, they arose in
5104-436: The eigenvectors are any nonzero scalar multiples of v λ = 1 = [ 1 − 1 ] , v λ = 3 = [ 1 1 ] . {\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {v} _{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.} If
5192-550: The eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like d d x {\displaystyle {\tfrac {d}{dx}}} , in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as d d x e λ x = λ e λ x . {\displaystyle {\frac {d}{dx}}e^{\lambda x}=\lambda e^{\lambda x}.} Alternatively,
5280-432: The entries of A are all algebraic numbers , which include the rationals, the eigenvalues must also be algebraic numbers. The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates , namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of
5368-443: The entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be irrational numbers even if all the entries of A are rational numbers or even if they are all integers. However, if
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#17328525060535456-549: The following. (In the list below, u , v and w are arbitrary elements of V , and a and b are arbitrary scalars in the field F .) The first four axioms mean that V is an abelian group under addition. An element of a specific vector space may have various nature; for example, it could be a sequence , a function , a polynomial or a matrix . Linear algebra is concerned with those properties of such objects that are common to all vector spaces. Linear maps are mappings between vector spaces that preserve
5544-402: The four arithmetic operations; thus the rational numbers Q are excluded, but the surd field is acceptable. For this reason, not every scalar product space is a normed vector space. When the requirement that the set of scalars form a field is relaxed so that it need only form a ring (so that, for example, the division of scalars need not be defined, or the scalars need not be commutative ),
5632-417: The induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces . More precisely, a linear subspace of a vector space V over a field F is a subset W of V such that u + v and a u are in W , for every u , v in W , and every a in F . (These conditions suffice for implying that W is a vector space.) For example, given
5720-849: The inequality γ A ( λ ) ≤ μ A ( λ ) {\displaystyle \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )} , consider how the definition of geometric multiplicity implies the existence of γ A ( λ ) {\displaystyle \gamma _{A}(\lambda )} orthonormal eigenvectors v 1 , … , v γ A ( λ ) {\displaystyle {\boldsymbol {v}}_{1},\,\ldots ,\,{\boldsymbol {v}}_{\gamma _{A}(\lambda )}} , such that A v k = λ v k {\displaystyle A{\boldsymbol {v}}_{k}=\lambda {\boldsymbol {v}}_{k}} . We can therefore find
5808-409: The largest integer k such that ( λ − λ i ) divides evenly that polynomial. Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues. Whereas equation ( 4 ) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of d terms each corresponding to
5896-442: The linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices. If the linear transformation is expressed in the form of an n by n matrix A , then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication A v = λ v , {\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,} where
5984-525: The linear transformation is applied to it: T v = λ v {\displaystyle T\mathbf {v} =\lambda \mathbf {v} } . The corresponding eigenvalue , characteristic value , or characteristic root is the multiplying factor λ {\displaystyle \lambda } (possibly negative). Geometrically, vectors are multi- dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates , stretches , or shears
6072-584: The linear transformation, and the associated eigenvector is the steady state of the system. Consider an n × n {\displaystyle n{\times }n} matrix A and a nonzero vector v {\displaystyle \mathbf {v} } of length n . {\displaystyle n.} If multiplying A with v {\displaystyle \mathbf {v} } (denoted by A v {\displaystyle A\mathbf {v} } ) simply scales v {\displaystyle \mathbf {v} } by
6160-492: The method of elimination, which was initially listed as an advancement in geodesy . In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix , which is Latin for womb . Linear algebra grew with ideas noted in the complex plane . For instance, two numbers w and z in C {\displaystyle \mathbb {C} } have
6248-993: The nullspace of ( A − λI ), also called the nullity of ( A − λI ), which relates to the dimension and rank of ( A − λI ) as γ A ( λ ) = n − rank ( A − λ I ) . {\displaystyle \gamma _{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).} Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n . 1 ≤ γ A ( λ ) ≤ μ A ( λ ) ≤ n {\displaystyle 1\leq \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )\leq n} To prove
6336-407: The only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i . A set of vectors that spans a vector space is called a spanning set or generating set . If a spanning set S is linearly dependent (that is not linearly independent), then some element w of S is in the span of the other elements of S , and the span would remain
6424-501: The operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers ). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on
6512-431: The other by elementary row and column operations . For a matrix representing a linear map from W to V , the row operations correspond to change of bases in V and the column operations correspond to change of bases in W . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from W to V , there are bases such that
6600-661: The real part of a quaternion: A vector space is defined as a set of vectors (additive abelian group ), a set of scalars ( field ), and a scalar multiplication operation that takes a scalar k and a vector v to form another vector k v . For example, in a coordinate space , the scalar multiplication k ( v 1 , v 2 , … , v n ) {\displaystyle k(v_{1},v_{2},\dots ,v_{n})} yields ( k v 1 , k v 2 , … , k v n ) {\displaystyle (kv_{1},kv_{2},\dots ,kv_{n})} . In
6688-424: The result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts. Two matrices that encode the same linear transformation in different bases are called similar . It can be proved that two matrices are similar if and only if one can transform one into
6776-406: The resulting more general algebraic structure is called a module . In this case the "scalars" may be complicated objects. For instance, if R is a ring, the vectors of the product space R can be made into a module with the n × n matrices with entries from R as the scalars. Another example comes from manifold theory , where the space of sections of the tangent bundle forms a module over
6864-707: The right-hand side is the product of n linear terms and this is the same as equation ( 4 ). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as 1 ≤ μ A ( λ i ) ≤ n , μ A = ∑ i = 1 d μ A ( λ i ) = n . {\displaystyle {\begin{aligned}1&\leq \mu _{A}(\lambda _{i})\leq n,\\\mu _{A}&=\sum _{i=1}^{d}\mu _{A}\left(\lambda _{i}\right)=n.\end{aligned}}} If μ A ( λ i ) = 1, then λ i
6952-411: The roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs. The spectrum of a matrix is the list of eigenvalues, repeated according to multiplicity; in an alternative notation
7040-419: The same if one were to remove w from S . One may continue to remove elements of S until getting a linearly independent spanning set . Such a linearly independent set that spans a vector space V is called a basis of V . The importance of bases lies in the fact that they are simultaneously minimal generating sets and maximal independent sets. More precisely, if S is a linearly independent set, and T
7128-416: The same vector space, a linear map T : V → V is also known as a linear operator on V . A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism . Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in
7216-524: The sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm . The study of those subsets of vector spaces that are in themselves vector spaces under
7304-414: The set of eigenvalues with their multiplicities. An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the spectral radius of the matrix. Let λ i be an eigenvalue of an n by n matrix A . The algebraic multiplicity μ A ( λ i ) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is,
7392-403: The study of quadratic forms and differential equations . In the 18th century, Leonhard Euler studied the rotational motion of a rigid body , and discovered the importance of the principal axes . Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix. In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify
7480-426: The vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices , or the language of linear transformations. The following section gives a more general viewpoint that also covers infinite-dimensional vector spaces . Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen-
7568-432: The vector-space structure. Given two vector spaces V and W over a field F , a linear map (also called, in some contexts, linear transformation or linear mapping) is a map that is compatible with addition and scalar multiplication, that is for any vectors u , v in V and scalar a in F . This implies that for any vectors u , v in V and scalars a , b in F , one has When V = W are
7656-417: The vectors upon which it acts. Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed. The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all
7744-488: Was "proper value", but the more distinctive term "eigenvalue" is the standard today. The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method . One of the most popular methods today, the QR algorithm , was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961. Eigenvalues and eigenvectors are often introduced to students in
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