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Topic: Positive definite

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Bilinear form

Equivalence class

Hessian matrix

Likelihood

Multivariate normal distribution

Jacobian

Lebesgue integration

Square root

Probability density function

Eigenvalue

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	Positive-definite matrix -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-13)
		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.
		Every positive definite matrix, has at least one (A number that when multiplied by itself equals a given number) square root matrix such that.
	www.absoluteastronomy.com /encyclopedia/p/po/positive-definite_matrix.htm (444 words)

	PlanetMath: positive definite
		The definiteness of a matrix is an important property that has use in many areas of mathematics and even physics.
		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.
	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.
		Positive definite vs positve definite form by iwnbap on 2003-04-10 02:58:52
		There seem to be two entries, one for "positive definite" the other for "positve definite form"; they don't seem to reference each other.
	planetmath.org /encyclopedia/Indefinite.html (126 words)

	Definite bilinear form - Wikipedia, the free encyclopedia
		In mathematics, a definite bilinear form B is one for which
		has a fixed sign (positive or negative) when it is not 0.
		To give a formal definition, let K be one of the fields R (real numbers) or C (complex numbers).
	www.wikipedia.org /wiki/Positive-definite (151 words)

	Learn more about Real number in the online encyclopedia. (Site not responding. Last check: 2007-10-13)
		Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.
		The first rigorous definition was given by Georg Cantor in 1871.
		Self-adjoint operatorss on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra.
	www.onlineencyclopedia.org /r/re/real_number.html (2010 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.
		So the failure of a matrix to be positive definite may indicate a problem with the input matrix.
		A not positive definite input covariance matrix may signal a perfect linear dependency of one variable on another.
	www.gsu.edu /~mkteer/npdmatri.html (2444 words)

	CATHOLIC ENCYCLOPEDIA: Life
		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.
		The definition applies to plants, animals, and man. The human soul, however, endowed with rationality is of a higher grade.
		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.
	www.newadvent.org /cathen/09238c.htm (5526 words)

	Positive-definite matrix
		(6) All the following matrices have positive determinant: the upper left 1-by-1 corner of M, the upper left 2-by-2 corner of M, the upper left 3-by-3 corner of M,..., and M itself.
		If M is positive definite and r > 0 is a real number, then rM is positive definite.
		Suppose K denotes the field R or C, V is a vector space over K, and B : V × V → K is a bilinear map which is Hermitian in the sense that B(x,y) is always the complex conjugate of B(y,x).
	www.sciencedaily.com /encyclopedia/positive_definite_matrix (373 words)

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

	Spectral Perturbation Bounds For Positive Definite Matrices - Mathias (ResearchIndex) (Site not responding. Last check: 2007-10-13)
		It was shown by Barlow and Demmel, and Demmel and Veseli'c that if one takes a component-wise approach one can prove much stronger bounds on i (H)= i (H +\DeltaH) and the components of the eigenvectors of H and H +\DeltaH than by using the standard norm-wise perturbation theory.
		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
	citeseer.ist.psu.edu /mathias97spectral.html (546 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPBS, DPBS, SPBCHS, ...
		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.
		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.
		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.
	www.nersc.gov /vendor_docs/ibm/essl/essl162.html (519 words)

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

	30. (Site not responding. Last check: 2007-10-13)
		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)

	Positive definite Matrices (Site not responding. Last check: 2007-10-13)
		Foerstner 95: A Metric for Comparing Symmetric Positive Definite Matrices...
		R help archive: [R] not positive definite D matrix in quadprog...
		Curve fit matrices are positive definite - Cholesky...
	www.scienceoxygen.com /math/251.html (275 words)

	cptsvx mitian positive definite tridiagonal matrix and X and B are N-by-NRHS ma... (Site not responding. Last check: 2007-10-13)
		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)

	Guide and Reference (Site not responding. Last check: 2007-10-13)
		A is the positive definite real symmetric or complex Hermitian matrix to be factored.
		Example 1: This example factors a 9 × 9 positive definite real symmetric matrix.
		Example 2: This example factors a 9 × 9 positive definite complex Hermitian matrix.
	csit1cwe.fsu.edu /extra_link/pessl/pessl249.html (983 words)

	[No title]
		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)

	Foerstner 95: A Metric for Comparing Symmetric Positive Definite Matrices (Site not responding. Last check: 2007-10-13)
		The note proposes a metric for comparing symmetric positive definite matrices.
		Symmetric positive definite matrices can be interpreted as covariance matrices.
		The metric is shown to be invariant with respect to common affine transformations of the reference coordinate system of the covariance matrices and to inversions of the matrices, thus also measures the distance between the corresponding weight matrices.
	www.ipb.uni-bonn.de /ipb/lit/abstracts95/foerstner95.metric.html (77 words)

	Positive Definite Matrices (Site not responding. Last check: 2007-10-13)
		A matrix is positive definite if it is Hermitian and
		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:
	www.eng.fsu.edu /~dommelen/courses/aim/aim99/topics/linalg/eigs/positive/index.html (54 words)

	A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation (Site not responding. Last check: 2007-10-13)
		In several recent papers, a one-sided iterative process for computing positive definite solutions of the nonlinear matrix equation
		In this paper, a two-sided iterative process for the same equation is investigated.
		is normal and satisfies an additional condition, the matrix equation has smallest and largest positive definite solutions.
	www.austms.org.au /Publ/Jamsb/V45P1/1860.html (128 words)

	Matrix Manual: Special Matrices (Site not responding. Last check: 2007-10-13)
		Such matrices are therefore excluded from the concept of definiteness.
		B is +ve definite iff the columns of B are linearly independent (B not necessarily square)
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
	www.psi.toronto.edu /matrix/special.html (2192 words)

	4.2 Positive Definite Linear Equations (Site not responding. Last check: 2007-10-13)
		Solves a positive definite real or complex set of 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
	www.ee.ucla.edu /~vandenbe/cvxopt/doc/e-pd-eqs.html (176 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.
		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.
		V. Arestov and E. Berdysheva, Turán's problem for positive definite functions with support in a hexagon, Proc.
	www.ams.org /tran/2004-356-11/S0002-9947-04-03502-0/home.html (1070 words)

	Equilibration Scale Factors for a Symmetric Positive Definite Matrix in Packed Storage (Site not responding. Last check: 2007-10-13)
		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 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)

	Matrix Reference Manual: Matrix Properties
		Such matrices are excluded from the concept of definiteness.
		notation section for definitions of sgn(), prod() and PERM().
		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.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/property.html (2323 words)

	Positive Definite Matrices
		These definitions are hard to check directly and you might as well forget them for all practical purposes.
		More useful in practice are the following properties, which hold when the matrix A is symmetric (that will be the case of interest to us), and which are easier to check.
		The matrix A is positive definite if all its principal minors
	mat.gsia.cmu.edu /QUANT/NOTES/chap1/node8.html (173 words)

	What is the nature of life?
		is still one of the two or three most difficult problems that face both scientist and philosopher, and notwithstanding the progress of knowledge during the past twenty-three hundred years, we do not seem to have advanced appreciably beyond the position of Aristotle in regard to the main issue.
		Where we would expect to find a definition of life, we find, instead, a list of questions, and then they offer a six-thousand-plus-word dissertation, giving us opinions and theories going all the way back to Aristotle.
		But what we don't find, is a clear definition of the word "life," and we don't find any answers to the questions they pose.
	www.choice101.com /74-what-is-life.html (1250 words)

	UNIVERSAL BINARY POSITIVE DEFINITE HERMITIAN LATTICES (Site not responding. Last check: 2007-10-13)
		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.
		A.G. Earnest and A. Khosravani determined universal binary Hermitian lattices when the imaginary quadratic fields have class number 1.
		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.
	math.la.asu.edu /~rmmc/rmj/VOL30-3/IWA/IWA.html (85 words)

	Sun Performance Library Reference: 1 - LAPACK Subroutines (Site not responding. Last check: 2007-10-13)
		Cholesky Factorization of a Symmetric Positive Definite Matrix in Packed Storage
		The subroutines described in this section compute a Cholesky factorization of a real symmetric (or Hermitian) positive definite matrix A in packed storage.
		C 4x4 symmetric positive definite coefficient matrix A
	www.crml.uab.edu /workshop/perflib/perflib_ref/chapter1.doc89.html (311 words)

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