Positive-definite - Factbites

Positive-definite - Factbites
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Topic: Positive-definite

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	PlanetMath: positive definite
		Thus the determinant of a positive definite matrix is positive, and a positive definite matrix is always invertible.
		This is version 6 of positive definite, born on 2002-02-15, modified 2004-06-17.
		C.R. Johnson, Positive definite matrices, American Mathematical Monthly, Vol.
	planetmath.org /encyclopedia/PositiveDefinite.html (237 words)

	PlanetMath: positive definite form
		This is version 2 of positive definite form, born on 2002-02-22, modified 2004-09-20.
		There seem to be two entries, one for "positive definite" the other for "positve definite form"; they don't seem to reference each other.
		Positive definite vs positve definite form by iwnbap on 2003-04-10 02:58:52
	planetmath.org /encyclopedia/Indefinite.html (126 words)

	Positive-definite matrix -- Facts, Info, and Encyclopedia article
		Every positive definite matrix, has at least one (A number that when multiplied by itself equals a given number) square root matrix such that.
		Every positive definite matrix is invertible and its inverse is also positive definite.
		If is positive definite and is a real number, then is positive definite.
	www.absoluteastronomy.com /encyclopedia/p/po/positive-definite_matrix.htm (444 words)

	Definite bilinear form - Wikipedia, the free encyclopedia
		If it is greater than or equal to zero, we say B is positive semidefinite.
		In mathematics, a definite bilinear form B is one for which
		has a fixed sign (positive or negative) when it is not 0.
	www.wikipedia.org /wiki/Positive-definite (151 words)

	Not Positive Definite Matrices--Causes and Cures
		First, the researcher may get a message saying that the input covariance or correlation matrix being analyzed is "not positive definite." Generalized least squares (GLS) estimation requires that the covariance or correlation matrix analyzed must be positive definite, and maximum likelihood (ML) estimation will also perform poorly in such situations.
		A not positive definite input covariance matrix may signal a perfect linear dependency of one variable on another.
		So the failure of a matrix to be positive definite may indicate a problem with the input matrix.
	www.gsu.edu /~mkteer/npdmatri.html (2444 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPTF and DPTF--Positive Definite Symmetric Tridiagonal Matrix Factorization
		The factorization of positive definite symmetric tridiagonal matrix A is computed using Gaussian elimination.
		For a description of how positive definite symmetric tridiagonal matrices are stored, see Positive Definite Symmetric Tridiagonal Matrix.
		However, these storage positions are required and may be overwritten during the computation.
	www.nersc.gov /vendor_docs/ibm/essl/essl168.html (465 words)

	CATHOLIC ENCYCLOPEDIA: Life
		The history of vitalism, which we have thus briefly outlined, shows how the advance of biological research and the trend of the best modern scientific thought is moving steadily back in the direction of that conception of life to be found in the scholastic philosophy, itself based on the teaching of Aristotle.
		The definition applies to plants, animals, and man. The human soul, however, endowed with rationality is of a higher grade.
		With him genuine scientific and philosophic treatment of the subject begins, and the position to which he advanced it is among the finest evidences of both his encyclopedic knowledge and his metaphysical genius.
	www.newadvent.org /cathen/09238c.htm (5526 words)

	Spectral Perturbation Bounds For Positive Definite Matrices - Mathias (ResearchIndex)
		Mathias, Spectral perturbation bounds for positive definite matrices, SIAM J. Matrix Anal.
		4 A relative perturbation bound for positive definite matrices - Mathias, Veseli'c - 1995
		Spectral Perturbation Bounds For Positive Definite Matrices - Mathias (ResearchIndex)
	citeseer.ist.psu.edu /mathias97spectral.html (546 words)

	30.
		that the values of the determinant of a positive definite matrix as well as of all its main diagonal elements are positive.
		The convergence condition, by which the matrix of the system of equations must be symmetric and positive definite, appears at first to be very restrictive.
		However, you can confirm without great effort that every arbitrary system of equations with a non-zero determinant can be replaced by a system with the same solution and a symmetric and positive definite matrix.
	kr.cs.ait.ac.th /~radok/math/mat3/m130.htm (1038 words)

	Critical Graphs for the Positive Definite Completion Problem
		The graph G is called critical for the positive definite completion problem if there is a critical partial positive definite matrix A, the graph of whose specified entries is G.
		A partial positive definite matrix A is "critical" if A has no positive definite completion, though every proper principal submatrix does.
		Among various matrix completion problems that have been considered in recent years, the positive definite completion problem seems to have received the most attention.
	epubs.siam.org /sam-bin/dbq/article/32457 (333 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPBS, DPBS, SPBCHS, and DPBCHS--Positive Definite Symmetric Band Matrix Solve
		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.
		If the factorization subroutine resulted in a nonpositive definite matrix, error 2104 for SPBF and DPBF or error 2115 for SPBCHF and DPBCHF, results of these subroutines may be unpredictable.
		The system Ax = b is solved for x, where A is a positive definite symmetric band matrix, stored in lower-band-packed storage mode, and x and b are vectors.
	www.nersc.gov /vendor_docs/ibm/essl/essl162.html (519 words)

	Guide and Reference
		A is the positive definite real symmetric or complex Hermitian matrix to be factored.
		Example 2: This example factors a 9 × 9 positive definite complex Hermitian matrix.
		Example 1: This example factors a 9 × 9 positive definite real symmetric matrix.
	csit1cwe.fsu.edu /extra_link/pessl/pessl249.html (983 words)

	4.2 Positive Definite Linear Equations
		Returns 0 if successful and a positive integer if the matrix is not positive definite.
		Solves a positive definite real or complex set of linear equations (4.1) given the Cholesky factorization computed by
		Solves a positive definite real or complex set of linear equations
	www.ee.ucla.edu /~vandenbe/cvxopt/doc/e-pd-eqs.html (176 words)

	Positive Definite Matrices
		Note: not all authors require the matrix to be Hermitian to be positive definite.
		One way to check positive definiteness of a matrix is to verify whether all eigenvalues are positive:
		A matrix is positive definite if it is Hermitian and
	www.eng.fsu.edu /~dommelen/courses/aim/aim99/topics/linalg/eigs/positive/index.html (54 words)

	cptsvx mitian positive definite tridiagonal matrix and X and B are N-by-NRHS ma...
		cptsvx mitian positive definite tridiagonal matrix and X and B are N-by-NRHS ma...
		CPTSVX uses the factorization A = LDL*H to compute the solution to a complex system of linear equations AX = B, where A is an N-by-N Hermi- tian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices.
		The factorization can also be regarded as having the form A = U*HDU. If the leading i-by-i principal minor is not positive* definite, then the routine returns with INFO = i.
	www.uni-kiel.de /rz/nvv/altix-doc/man_html/man3/cptsvx.3s.html (592 words)

	chua.html
		The first result is the polynomial-representability of homogeneous cones by positive definite cone.
		Using $T$-algebras, a class of non-associative algebras invented by Vinberg in the early 1960's, I proved that each homogeneous cone can be expressed as the intersection of a cone of positive definite matrices and a linear subspace.
		By using second-order cone approximations to positive cone, I developed a new primal-dual interior-point algorithm for semi-definite programming that runs in $O(\sqrt{n})$ iterations, the best bound known.
	www.math.cmu.edu /users/nw0z/abstracts/chua.html (146 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)

	Positive-definite function - Wikipedia, the free encyclopedia
		Positive-definiteness arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positive-definite is a necessary condition on f, for it to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
		The converse result is Bochner's theorem, stating that a continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.
		This result generalises to the context of Pontryagin duality, with positive-definite functions defined on any locally compact abelian topological group.
	en.wikipedia.org /wiki/Positive-definite_function (197 words)

	Transactions of the American Mathematical Society
		Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support.
		V. Arestov and E. Berdysheva, Turán's problem for positive definite functions with support in a hexagon, Proc.
		Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions.
	www.ams.org /tran/2004-356-11/S0002-9947-04-03502-0/home.html (1070 words)

	UNIVERSAL BINARY POSITIVE DEFINITE HERMITIAN LATTICES
		We will determine all universal integral lattices on binary positive definite Hermitian spaces over arbitrary imaginary quadratic fields, where a positive definite lattice is said to be `universal' if it represents all positive rational integers.
		In this paper we will extend the result to the case of fields with arbitrary class numbers and obtain nine new universal binary Hermitian lattices up to equivalence, including nonfree lattices.
		A.G. Earnest and A. Khosravani determined universal binary Hermitian lattices when the imaginary quadratic fields have class number 1.
	math.la.asu.edu /~rmmc/rmj/VOL30-3/IWA/IWA.html (85 words)

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

	Matrix Manual: Special Matrices
		B is +ve definite iff the columns of B are linearly independent (B not necessarily square)
		Such matrices are therefore excluded from the concept of definiteness.
		If A is +ve definite then INV(A) exists and is +ve definite.
	www.psi.toronto.edu /matrix/special.html (2192 words)

	Toeplitz Positive Definite
		In this paper it is shown that a partial Toeplitz positive definite matrix has a Toeplize positive definite completion if and only if the diagonals for the specified entries are 0, t, 2t,
		This gives a positive answer to the conjecture posed by C. Johnson.
	www.math.cuhk.edu.hk /conference/may2002/abstract/he_m.html (72 words)

	Positive Definite The story of our daughter
		It's hard to see, but it's definitely there.
		She was chomping on my thumb and I felt it scraping across the lines of my fingerprint.
	posdef.net (562 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.
		Such matrices are excluded from the concept of definiteness.
		notation section for definitions of sgn(), prod() and PERM().
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/property.html (2323 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.
		Equilibration Scale Factors for a Symmetric Positive Definite Matrix in Packed Storage
		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)

	A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation
		is normal and satisfies an additional condition, the matrix equation has smallest and largest positive definite solutions.
		In several recent papers, a one-sided iterative process for computing positive definite solutions of the nonlinear matrix equation
		ANZIAM J. A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation
	www.austms.org.au /Publ/Jamsb/V45P1/1860.html (128 words)

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In the News (Fri 25 Dec 09)