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Hermitian matrix - Wikipedia, the free encyclopedia A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j: A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite; if they are all non-negative, then the matrix is positive semidefinite. en.wikipedia.org /wiki/Hermitian_matrix   (426 words)

Art Fresh : Article 'Matrix' Similarity matrix Skew-Hermitian matrix - a square matrix which is equal to the negative of its conjugate transpose, A* = −A. Skew-symmetric matrix - a matrix which is equal to the negative of its transpose, AT = −A. Sparse matrix - containing mostly zeros Square matrix - an n by n matrix. Unitary matrix - a square matrix whose inverse is equal to its conjugate transpose, A−1 = A*. Orthogonal matrix - a matrix whose inverse is equal to its transpose, A−1 = AT. www.art-fresh.net /DisplayArticle45184.html   (1636 words)

Skew-Hermitian matrix - Wikipedia, the free encyclopedia All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, ie. In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A An arbitrary matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B: en.wikipedia.org /wiki/Skew-Hermitian_matrix   (178 words)

Diagonalisable matrices Note that a symmetric matrix is always Hermitian, a skew-symmetric matrix is always skew-Hermitian and an orthogonal matrix is always unitary. is a diagonal matrix with the eigenvalues of complex matrix is unitarily similar to an upper triangular matrix. home.iitk.ac.in /~arlal/book/nptel/mth102/node57.html   (446 words)

Example 5: A Hermitian matrix The diagonal elements of a skew Hermitian matrix are pure imaginary or zero. => B is skew Hermitian (a skew Hermite matrix). The diagonal elements of a Hermitian matrix are real. www.ee.oulu.fi /~mpa/matreng/eem1_3-5.htm   (42 words)

Linear Algebra Glossary The inverse of a (nonsingular) circulant matrix is a circulant matrix. The transpose of a circulant matrix is a circulant matrix. A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix. www.math.iastate.edu /burkardt/papers/linear_glossary.html   (13535 words)

Self-adjoint operator - Wikipedia, the free encyclopedia = H − iΓ is the sum of an Hermitian H and a skew-Hermitian (see skew-Hermitian matrix) operator − iΓ, one defines the biorthogonal basis set On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. If H Hermitian (the name for self-adjoint in the physics literature) and f is a Borel function, www.wikipedia.org /wiki/Hermitian_operator   (2185 words)

Real/Imaginary Eigen Values An analogous q*d/q proof shows m is skew hermitian iff it has an orthogonal eigen basis and imaginary eigen values. A similar proof shows the inverse of a skew hermitian matrix is skew hermitian. An analogous proof shows the eigen values of a skew hermitian operator are pure imaginary. www.mathreference.com /la-sim,realim.html   (811 words)

Citebase - Solution of the congruence problem for arbitrary hermitian and skew-hermitian matrices over polynomial rings Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to the direct sum of 1 by 1 matrices and 2 by 2 matrices with zero diagonal. Morevover, we show that if two n by n hermitian or skew-hermitian matrices have the same invariant factors, then they are congruent. Authors: Djokovic, D. echtman, F. We equip the complex polynomial algebra C[t] with the involution which is the identity on C and sends t to -t. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0206104   (306 words)

Canonical Forms for Doubly Structured Matrices and Pencils - Mehl, Mehrmann, Xu (ResearchIndex) Abstract: In this paper we derive canonical forms under structure preserving equivalence transformations for matrices and matrix pencils that have a multiple structure, which is either an H-selfadjoint or H-skew-adjoint structure, where the matrix H is a complex nonsingular Hermitian or skew-Hermitian matrix. 12 Solution of large matrix equations which occur in response t.. Matrices and pencils of such multiple structures arise for example in quantum chemistry in Hartree-Fock models or random phase approximation. citeseer.ist.psu.edu /mehl00canonical.html   (446 words)

Mathematical Methods (10/24.539) Overview of Linear Algebra Some Special Matrices Note that hermitian, skew-hermitian, and unitary matrices are, in general, complex matrices. The eigenvalues of a hermitian (or real symmetric) matrix are real. Also note that the magnitude of each eigenvalue for the unitary matrix is indeed unity. gershwin.ens.fr /vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s3/s3spm/s3spm.html   (396 words)

SIMAX Volume 24 Issue 3 We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. epubs.siam.org /sam-bin/dbq/article/39545   (244 words)

skewpart.m % It is the nearest skew-symmetric (skew-Hermitian) matrix to A in % both the 2- and the Frobenius norms. www.cs.unc.edu /~smp/COMP205/Program/Matrix/skewpart.m   (40 words)

Math Notes skew-symmetric matrix is a special case of skew-symmetric hermitian matrix such that they share the same properties. Given a centered data matrix X, X*X' is the dimension covarance matrix (ignore the factor 1/n) according to its definition (m x m) that measures the variance of data to the mean. Mathematics analysis has proved that the principal components of the gram matrix of the centered data are the continuous solutions to the discrete cluster membership indicators for k-means clustering. jekky-math.blogspot.com   (674 words)

Current Projects of Barry Mazur This organization consists of a single skew-Hermitian matrix with entries in the Iwasawa algebra associated to L/K, the maximal Z The next four articles are about the construction (for triples (p,K,E) satisfying some hypotheses, where p is a prime number, K is a number field, and E is an elliptic curve over K) of what we call an organization of the arithmetic of (p,K,E). abel.math.harvard.edu /~mazur/projects.html   (1182 words)

iqexpand.com Look for Skew-hermitian matrix in the Commons, our repository for free images, music, sound, and video. Look for Skew-hermitian matrix in Wiktionary, our sister dictionary project. is the sum of an Hermitian H and a skew-Hermitian (see skew-hermitian_matrix.iqexpand.com /index.php?...&action=edit   (137 words)

SIMAX Volume 26 Issue 2 In this paper we derive bounds on the eigenvalues of the preconditioned matrix that arises in the solution of saddle point problems when the Hermitian and skew-Hermitian splitting preconditioner is employed. Spectral Properties of the Hermitian and Skew-Hermitian Splitting Preconditioner for Saddle Point Problems We also give sufficient conditions for the eigenvalues to be real. dx.doi.org /10.1137/S0895479803434926   (141 words)

SISC Volume 26 Issue 3 Theoretical analysis shows that the PSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrix and independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. When we specialize the PSS to block triangular (or triangular) and skew-Hermitian splitting (BTSS or TSS), the PSS method naturally leads to a BTSS or TSS iteration method, which may be more practical and efficient than the HSS and NSS iteration methods. non-Hermitian matrix, positive-definite matrix, triangular matrix, block triangular matrix, Hermitian and skew-Hermitian splitting, splitting iteration method epubs.siam.org /sam-bin/dbq/article/42811   (230 words)

SIMAX Volume 24 Issue 3 eigenvalue, backward error, condition number, eigenvector, symmetric matrix, Hermitian matrix, skew-symmetric matrix, skew-Hermitian matrix, symplectic matrix, conjugate symplectic matrix, Hamiltonian matrix epubs.siam.org /sam-bin/dbq/article/39995   (178 words)

Skew-Hermitian matrix - Wikipedia, the free encyclopedia All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, ie. In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A An arbitrary matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B: en.wikipedia.org /wiki/Skew-Hermitian_matrix   (178 words)

Matrix Manual: Special Matrices [3#3]: All 3#3 skew-symmetric matrices have the form SKEW ( a) = [0 - a Hermitian matrices are closed under addition, multiplication by a scalar, raising to an integer power, and (if non-singular) inversion. If A has distinct eigenvalues and rank( P)=1 then if the eigenvalues are arranged in descending order, the corresponding eigenvectors will be alternately symmetric and skew-symmetric with the first one being symmetric or skew-symmetric according to whether the non-zero eigenvalue of P is positive or negative. www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html   (178 words)

Real/Imaginary Eigen Values An analogous q*d/q proof shows m is skew hermitian iff it has an orthogonal eigen basis and imaginary eigen values. A similar proof shows the inverse of a skew hermitian matrix is skew hermitian. We already know that a hermitian matrix is normal, and has orthogonal eigen vectors, but let's prove this for general hermitian operators, even those that cannot be implemented by a matrix. www.mathreference.com /la-sim,realim.html   (811 words)

PlanetMath: Cayley's parameterization of orthogonal matrices Hermitian operator or study symmetric extensions of a symmetric operator by first performing a Cayley transform and dealing with the resulting bounded operator. Hermitian matrix, the Cayley transform is often written as follows when dealing with unitary matrices: The proof in this case is substantially the same as was presented above; all one has to do is replace matrix transposition with Hermitian conjugation. planetmath.org /encyclopedia/CayleysParameterizationOfOrthogonalMatrices.html   (572 words)

Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems: SIAM Journal on Matrix Analysis and Applications Vol. We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. epubs.siam.org /sam-bin/dbq/article/39545   (572 words)

Mathematical Methods (10/24.539) Overview of Linear Algebra Some Special Matrices Note that hermitian, skew-hermitian, and unitary matrices are, in general, complex matrices. In particular, of interest here are three classes of matrices-- Hermitian, Skew-Hermitian, and Unitary Matrices. The eigenvalues of a hermitian (or real symmetric) matrix are real. gershwin.ens.fr /vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s3/s3spm/s3spm.html   (572 words)

Skew-Hermitian matrix - Wikipedia, the free encyclopedia An arbitrary matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B: If A is skew-Hermitian, then iA is Hermitian This page was last modified 04:00, 23 September 2005. en.wikipedia.org /wiki/Skew-Hermitian_matrix   (572 words)

Notation of a Matrix and Operations with Matrices as a sum of a symmetric and a skew-symmetric matrix. can be expressed as a sum of a symmetric matrix and a skew-symmetric matrix. (i = 1 : m) of the transpose of matrix A ("pasting'' the matrices of the column-vectors or of the transposed row-vectors) www.cs.ut.ee /~toomas_l/linalg/lin1/node12.html   (172 words)

Diagonalisable matrices Note that a symmetric matrix is always Hermitian, a skew-symmetric matrix is always skew-Hermitian and an orthogonal matrix is always unitary. is a diagonal matrix with the eigenvalues of complex matrix is unitarily similar to an upper triangular matrix. home.iitk.ac.in /~arlal/book/nptel/la/node57.html   (446 words)

mmio.f Valid inputs: c c symmetric c hermitian c skew-symmetric c general c c rows integer in Number of rows in matrix. On return: c c symmetric c hermitian c skew-symmetric c general c c rows integer out Number of rows in matrix. math.nist.gov /~KRemington/mmio/mmio.f   (1652 words)

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