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Positive-definite matrix - Wikipedia, the free encyclopedia Every positive definite matrix is invertible and its inverse is also positive definite. In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. Every positive definite matrix M, has at least one square root matrix N such that N en.wikipedia.org /wiki/Positive-definite_matrix   (465 words)

Matrix Reference Manual: Matrix Properties The inertia of an m#m square matrix is the triple (p,n,z) where p+n+z=m and p, n and z are respectively the number of eigenvalues, counting multiplicities, with positive, negative and zero real parts. The determinant of an orthogonal matrix is ±1. The determinant of a permutation matrix equals the signature of the column permutation. www.ee.ic.ac.uk /hp/staff/dmb/matrix/property.html   (2389 words)

Schur complement - Wikipedia, the free encyclopedia If M is a positive definite symmetric matrix, then so is the Schur complement of D in M. If we take the matrix V above to be, not a variance of a random vector, but a sample variance, then it may have a Wishart distribution. In linear algebra and the theory of matrices, the Schur complement (named after Issai Schur) of a block of a matrix within the larger matrix is defined as follows. www.wikipedia.org /wiki/Schur_complement   (361 words)

Guide and Reference A is the positive definite symmetric band matrix to be factored. If A is not positive definite, this subroutine uses the Matrix A, output from PBTRF, must be passed, unchanged, to the solve subroutine PBTRS. csit1cwe.fsu.edu /extra_link/pessl/pessl253.html   (1074 words)

Not Positive Definite Matrices--Causes and Cures ), the model-implied covariance matrix, is not positive definite. If a covariance or correlation matrix is not positive definite, then one or more of its eigenvalues will be negative. In addition, one interpretation of the determinant of a covariance or correlation matrix is as a measure of "generalized variance." Since negative variances are undefined, and since zero variances apply only to constants, it is troubling when a covariance or correlation matrix fails to have a positive determinant. www.gsu.edu /~mkteer/npdmatri.html   (2444 words)

Positive Definite Systems is positive definite, then all the submatrices of the matrix A obtained by deleting the rows and columns of the matrix A with the same numbers are positive definite and all the elements on the leading diagonal of the matrix are positive. is a positive definite matrix and the column vectors of the matrix is an upper triangular matrix with the unit diagonal, the matrices B and D have the same leading diagonal and the elements on it must be positive, provided that B is positive definite. www.cs.ut.ee /~toomas_l/linalg/lin2/node24.html   (538 words)

Vandermonde matrix - Wikipedia, the free encyclopedia In that case one may use a generalization called confluent Vandermonde matrices, which makes the matrix positive definite while retaining most properties. A commonly known special Vandermonde matrix is the discrete Fourier transform matrix. The determinant of an n × n Vandermonde matrix can be expressed as: en.wikipedia.org /wiki/Vandermonde_matrix   (440 words)

Positive definite Matrices R help archive: [R] not positive definite D matrix in quadprog... Curve fit matrices are positive definite - Cholesky... Foerstner 95: A Metric for Comparing Symmetric Positive Definite Matrices... www.scienceoxygen.com /math/251.html   (275 words)

Local Storage Scheme for Narrow Band Matrices Figures 4.11 and 4.12 also illustrate that the leading dimension of the local arrays containing the coefficient matrix must be at least BW+1 for the symmetric positive definite narrow band linear solver. matrix of right-hand-side vectors B (for example, used in PxGBTRS    , PxDBTRS    , and PxPBTRS    ) is assumed to be a dense matrix distributed in a block-row manner across the process grid. Thus, consecutive blocks of rows of the matrix B are assigned to successive processes in the process grid, as described in section 4.4.1. www.netlib.org /scalapack/slug/node84.html   (561 words)

R help archive: [R] not positive definite D matrix in quadprog One of them is to use the quadprog package in R. When I check positive definiteness of the D matrix, I get that one of the eigenvalues is negative of order 10^(-8). matrix D in quadratic function is not positive definite! I try to use solve.QP, but I get an error message that matrix D in quadratic function is not positive definite. tolstoy.newcastle.edu.au /R/help/04/09/3165.html   (403 words)

R: Class pdMat, positive-definite matrices : the logarithm of the determinant of the factor of the positive-definite matrix represented by the object. : the dimensions of the positive-definite matrix represented by the object. : Extract the correlation matrix corresponding to the positive-definite matrix represented by the object. www.stat.ucl.ac.be /ISdidactique/Rhelp/library/lme4/html/pdMat-class.html   (316 words)

OptiVec: MFsym_sqrt It uses eigenvalues and eigenvectors and fails whenever a negative eigenvalue is encountered, indicating an input matrix which is not positive definite. The square-root of a symmetric, positive definite matrix is calculated, so that MB * MB = MA is satisfied. Please note that non-symmetric matrices or matrices which are not positive definite may or may not have existing square-roots, too. www.optivec.com /matfuncs/sqrt.htm   (162 words)

SIMAX Volume 11 Issue 1 Parallel iterative methods are studied, and the focus is on linear algebraic systems whose matrix is symmetric and positive definite. The parallel iteration matrix is then formed by a weighted sum of iteration matrices that are associated with splittings of the matrix corresponding to the blocks. When the blocks are from a matrix in dissection form, it can be shown under suitable conditions that the parallel algorithm is convergent. locus.siam.org /SIMAX/volume-11/art_0611004.html   (166 words)

Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPBF, DPBF, SPBCHF, and DPBCHF--Positive Definite Symmetric Band Matrix Factorization The positive definite symmetric band matrix A, stored in lower-band-packed storage mode, is factored using Gaussian elimination in SPBF and DPBF and Cholesky factorization in SPBCHF and DPBCHF. For a description of how a positive definite symmetric band matrix is stored in lower-band-packed storage mode in an array, see Positive Definite Symmetric Band Matrix. Matrix A is not positive definite (for SPBF and DPBF). www.nersc.gov /vendor_docs/ibm/essl/essl161.html   (778 words)

R: General Positive-Definite Matrix is a numeric vector, it is assumed to represent the unrestricted coefficients of the matrix-logarithm parametrization of the underlying positive-definite matrix. It must have length equal to the dimension of the underlying positive-definite matrix and unreplicated elements. an optional vector of character strings specifying the row/column names for the matrix represented by object. www.stat.umn.edu /R/library/nlme/html/pdSymm.html   (249 words)

Information Technology Services When I run my data I get an error message that states that my covariance matrix is not positive definite. Alternatively, you may be inputting a correlation or covariance matrix which is based on incorrectly read raw data values via PRELIS, SPSS, or another program which has the capability to convert raw data into correlation or covariance matrix form. The difference between the number of estimable parameters and the number of estimated parameters is the number of degrees of freedom used in the chi-square test by LISREL (the first chi-square test to appear on the printout, not the Independence Model chi-square test). www.utexas.edu /cc/faqs/stat/lisrel/lisrel3.html   (378 words)

U.S. Pregrant 20040139137 - Method and apparatus for determining an inverse square root of a given positive-definite hermitian matrix Generally, a method and apparatus are provided for computing a matrix inverse square root of a given positive-definite Hermitian matrix, K. The disclosed technique for computing an inverse square root of a matrix may be implemented, for example, by the noise whitener of a MIMO receiver. The disclosed iterative algorithm determines the matrix, Q, given the covariance matrix, K. The disclosed matrix inverse square root determination process initially establishes an initial matrix, Q0, by multiplying an identity matrix by a scalar value and then continues to iterate and compute another value of the matrix, Qn+1, until a convergence threshold is satisfied. Conventional noise whitening algorithms whiten a non-white vector, X, by applying a matrix, Q, to X, such that the resulting vector, Y, equal to Q?X, is a white vector. cxp.paterra.com /uspregrant20040139137.html   (277 words)

The Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix and Rational Hermitian Interpolation - Mackens, Voss (ResearchIndex) Abstract: A novel method for computing the minimal eigenvalue of a symmetric positive definite Toeplitz matrix is presented. Mackens, W. and Voss, H. (1997): The minimum eigenvalue of a symmetric positivedefinite Toeplitz matrix and rational Hermitian interpolation. 5 Bounds on the extreme eigenvalues of positive definite Toepl.. citeseer.ist.psu.edu /mackens97minimum.html   (615 words)

Positive Definite Matrices A square matrix A is positive definite if If the determinants are all nonnegative, then the matrix is positive semidefinite, The ith principal minor of A is the matrix mat.gsia.cmu.edu /QUANT/NOTES/chap1/node8.html   (173 words)

dptsvx ric positive definite tridiagonal matrix and X and B are N-by-NRHS ma... dptsvx ric positive definite tridiagonal matrix and X and B are N-by-NRHS ma... If FACT = ’N’, the matrix A is factored as A = L*D*L**T, where L is a unit lower bidiagonal matrix and D is diagonal. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. www.uni-kiel.de /rz/nvv/altix-doc/man_html/man3/dptsvx.3s.html   (622 words)

Reciprocal Condition Number of a Cholesky-Factored Symmetric Positive Definite Matrix in Packed Storage The subroutines described in this section estimate the reciprocal condition number of a real symmetric (or Hermitian) positive definite matrix A in packed storage, which has been Cholesky-factored by xPPTRF. The 1-norm or infinity-norm of matrix A. xRCOND On exit, the estimated reciprocal of the condition number of the matrix A where the reciprocal condition number of A is defined to be 1 / (ANORM x gata.matapl.uv.es /~mulet/cursos/cca/lapack/chapter1-85.html   (323 words)

Error 1314. Not a symmetric positive definite matrix. You have used the Cholesky method with a matrix that is not a symmetric positive definite matrix. Use error trapping to detect whether the matrix is a symmetric positive definite matrix while executing Cholesky method. This error may occur in the following situations: www.bluebit-software.com /matrix/version_31/Error1314.htm   (43 words)

Introduction to Econometrics - Matrix Algebra A special diagonal matrix is a matrix where all diagonal elements are equal to one (identity matrix denoted I). The determinant of an orthogonal matrix is equal to 1 or -1. The rank of a (m*n) matrix is equal to the rank of the largest sub matrix with a determinant different from zero where the determinant of a matrix is defined by www.resacorp.com /matrix_algebra.htm   (1849 words)

Equilibration Scale Factors for a Symmetric Positive Definite Matrix in Packed Storage The subroutines described in this section compute equilibration scale factors for a real symmetric (or Hermitian) positive definite matrix A in packed storage. Upper or lower triangle of the matrix A. The dimension of xA is (N x N + N) / 2. Equilibration is intended to reduce the condition number of A with respect to the 2-norm. gata.matapl.uv.es /~mulet/cursos/cca/lapack/chapter1-86.html   (439 words)

C Programming: random sparse symmetric positive definite matrix Symmetric is easy; positive definiteness is a little tougher, though if you look at the definition, along with one of the conditions equivalent to being positive definite, how to generate them recurring on the dimension of the matrix should be pretty clear. I need to generate a random sparse symmetric positive definite matrix of the dimension NxN (N rows and N columns). C Programming: random sparse symmetric positive definite matrix www.experts-exchange.com /Q_20390206.html   (385 words)

Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A * X-2 A = I Based on them, some properties of the coefficient matrix A are presented when the matrix equation has a positive definite solution. Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A * X-2 A = I Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A www.pku.edu.cn /academic/xb/2005/_2005e108.html   (102 words)

Refined Solution to a Linear System in a Cholesky-Factored Symmetric Positive Definite Matrix Upper or lower triangle of the matrix A. Leading dimension of the array A as specified in a dimension or type statement. The NxNRHS right-hand side matrix B. Leading dimension of the array B as specified in a dimension or type statement. On entry, the NxNRHS solution matrix X. On exit, the refined NxNRHS solution matrix X. Leading dimension of the array X as specified in a dimension or type statement. gata.uv.es /~mulet/cursos/cca/lapack/chapter1-79.html   (492 words)

Sun Performance Library Reference: 1 - LAPACK Subroutines Cholesky Factorization of a Symmetric Positive Definite Matrix in Packed Storage C 4x4 symmetric positive definite coefficient matrix A The leading minor of order i of A, where i = INFO, is not positive definite. www.crml.uab.edu /workshop/perflib/perflib_ref/chapter1.doc89.html   (311 words)

Computation of the Smallest Even and Odd Eigenvalues of a Symmetric Positive-Definite Toeplitz Matrix We propose an algorithm to compute the smallest even and odd eigenvalues of a real symmetric positive-definite Toeplitz matrix, which is based on the factorization of the characteristic polynomial into an even and an odd polynomial. Computation of the Smallest Even and Odd Eigenvalues of a Symmetric Positive-Definite Toeplitz Matrix: SIAM Journal on Matrix Analysis and Applications Vol. Newton's method is used to compute the smallest even and odd eigenvalues as the smallest roots of the even and odd characteristic polynomials, respectively. dx.doi.org /10.1137/S0895479803423354   (153 words)

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