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Modal Matrix
In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors. Specifically the modal matrix M for the matrix A is the ''n'' × ''n'' matrix formed with the eigenvectors of A as columns in M. It is utilized in the similarity transformation : D = M^AM, where D is an ''n'' × ''n'' diagonal matrix with the eigenvalues of A on the main diagonal of D and zeros elsewhere. The matrix D is called the spectral matrix for A. The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in M. Example The matrix :A = \begin 3 & 2 & 0 \\ 2 & 0 & 0 \\ 1 & 0 & 2 \end has eigenvalues and corresponding eigenvectors : \lambda_1 = -1, \quad \, \mathbf b_1 = \left( -3, 6, 1 \right) , : \lambda_2 = 2, \qquad \mathbf b_2 = \left( 0, 0, 1 \right) , : \lambda_3 = 4, \qquad \mathbf b_3 = \left( 2, 1, 1 \right) . A diagonal matrix D, similar to A is :D = \begin - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra 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 spaces of functions. 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 is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) For a finite-dimensional vector space a linear map T:V\to V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix representation T = PDP^ as above, then the column vectors of P form a basis consisting of eigenvectors of and the diagonal entries of D are the corresponding eigenvalues of with respect to this eigenvector basis, A is represented by Diagonalization is the process of finding the above P and Diagonalizable matrices and maps are especially easy for computations, once their eigenvalues and eigenvectors are known. One can raise a diagonal matrix D to a power by simply raising the diagonal entries to that power, and the determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Eigenvalues And Eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix Similarity
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with being the change of basis matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in . Motivating example When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end\right/math>. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with ''n'' columns and ''n'' rows is diagonal if \forall i,j \in \, i \ne j \implies d_ = 0. However, the main diagonal entries are unrestricted. The term ''diagonal matrix'' ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix Similarity
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with being the change of basis matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in . Motivating example When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Canonical Basis
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: * In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. * In a polynomial ring, it refers to its standard basis given by the monomials, (X^i)_i. * For finite extension fields, it means the polynomial basis. * In linear algebra, it refers to a set of ''n'' linearly independent generalized eigenvectors of an ''n''×''n'' matrix A, if the set is composed entirely of Jordan chains. * In representation theory, it refers to the basis of the quantum groups introduced by Lusztig. Representation theory The canonical basis for the irreducible representations of a quantized enveloping algebra of type ADE and also for the plus part of that algebra was introduced by Lusztig by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Jordan Chain
In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space; let \phi be a linear map in , the set of all linear maps from V into itself; and let A be the matrix representation of \phi with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of A that form a complete basis for V. That is, the matrix A may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue \lambda_i is greater than its geometric multiplicity (the nullity of the matrix (A-\lambda_i I), or the dimension of its nullspace). In this case, \lambda_i is called a defective eigenvalue and A is called a defective matrix. A generalized eigenvector x_i corresponding to \lambda_i, together with the matrix (A-\lambda_i I) generate a Jordan chain of linearly ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Generalized Eigenvector
In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space; let \phi be a linear map in , the set of all linear maps from V into itself; and let A be the matrix representation of \phi with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of A that form a complete basis for V. That is, the matrix A may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue \lambda_i is greater than its geometric multiplicity (the nullity of the matrix (A-\lambda_i I), or the dimension of its nullspace). In this case, \lambda_i is called a defective eigenvalue and A is called a defective matrix. A generalized eigenvector x_i corresponding to \lambda_i, together with the matrix (A-\lambda_i I) generate a Jordan chain of linearly indep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Jordan Normal Form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. Let ''V'' be a vector space over a field ''K''. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in ''K'', or equivalently if the characteristic polynomial of the operator splits into linear factors over ''K''. This condition is always satisfied if ''K'' is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |