Defective Matrix
In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n \times n matrix is defective if and only if it does not have n linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. An n \times n defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues \lambda with algebraic multiplicity m > 1 (that is, they are multiple roots of the characteristic polynomial), but fewer than m linearly independent eigenvectors associated with \lambda. If the algebraic multiplicity of \lambda exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors ... [...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 matrix (mathematics), 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 line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, 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 a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Johns Hopkins University Press
Johns Hopkins University Press (also referred to as JHU Press or JHUP) is the publishing division of Johns Hopkins University. It was founded in 1878 and is the oldest continuously running university press in the United States. The press publishes books and journals, and operates other divisions including fulfillment and electronic databases. Its headquarters are in Charles Village section of Baltimore, Maryland Maryland ( ) is a U.S. state, state in the Mid-Atlantic (United States), Mid-Atlantic region of the United States. It borders the states of Virginia to its south, West Virginia to its west, Pennsylvania to its north, and Delaware to its east .... In 2017, after the retirement of Kathleen Keane, who is credited with modernizing JHU Press for the digital age, the university appointed new director Barbara Pope. Overview Daniel Coit Gilman, the first president of Johns Hopkins University, inaugurated the press in 1878. The press began as the university's Public ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Matrix Diagonalization
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not unique.) This property exists for any linear map: 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 A = 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, T is represented by Diagonalization is the process of finding the above P and and makes many subsequent computations easier. One can raise a diagonal matrix D to a power by simply raising the diagonal entries to that power. The determinant of a diagonal matrix is ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Jordan Normal Form
\begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ \hphantom\lambda_n 1\hphantom\\ \hphantom\hphantom\lambda_n \end Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number ''λi'' on its main diagonal, and 1s directly above the main diagonal. The ''λi''s are the eigenvalues of the matrix; they need not be distinct. In linear algebra, a Jordan normal form, also known as a Jordan canonical form, 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 ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Jordan Matrix
In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form: \begin \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & 0 & \lambda \end . Definition Every Jordan block is specified by its dimension ''n'' and its eigenvalue \lambda\in R, and is denoted as . It is an n\times n matrix of zeroes everywhere except for the diagonal, which is filled with \lambda and for the superdiagonal, which is composed of ones. Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This square matrix, consisting of diagonal blocks, can be compactly indicated as J_\oplus \cdots \oplus J_ or \mathrm\le ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Normal Matrix
In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix satisfying the equation is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus A = U D U^* and A^* = U D^* U^*where D is a diagonal matrix whose diagonal values are in general complex. The left and right singular vectors in the singular value decomposition of a normal matrix A = U D V^* dif ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Unitary Matrix
In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (), so the equation above is written U^\dagger U = UU^\dagger = I. A complex matrix is special unitary if it is unitary and its matrix determinant equals . For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. Properties For any unitary matrix of finite size, the following hold: * Given two complex vectors and , multiplication by preserves their inner product; that is, . * is normal (U^* U = UU^*). * is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Geometric Multiplicity
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The e ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |