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POSITIVE DEFINITE MATRIX PROBLEM — SOLUTIONS
"... Abstract. The selfimposed rule of the CauchySchwarz Master Class was to keep matrix algebra to a bare minimum. This decision was made to impose a discipline of simplicity, but many babies were thrown out with the bath water. Here is one that is certainly simple enough to have been include, even as ..."
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as a warmup problem. It’s also useful. Problem: Give a necessary and sufficient condition on α and β in order that T 2 + αT + βI be positive definite for each selfadjoint matrix T. Solution: To find a necessary condition, take T to be the 1dimensional identity matrix that takes 1 to x. We then need
Computing the Logarithm of a Symmetric Positive Definite Matrix
 Appl. Numer. Math
"... A numerical method for computing the logarithm of a symmetric positive definite matrix is developed in this paper. It is based on reducing the original matrix to a tridiagonal matrix by orthogonal similarity transformations and applying Pad'e approximations to the logarithm of the tridiagonal m ..."
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Cited by 3 (2 self)
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A numerical method for computing the logarithm of a symmetric positive definite matrix is developed in this paper. It is based on reducing the original matrix to a tridiagonal matrix by orthogonal similarity transformations and applying Pad'e approximations to the logarithm of the tridiagonal
APPROXIMATING THE INVERSE OF A SYMMETRIC POSITIVE DEFINITE MATRIX
"... It is shown for an n \Theta n symmetric positive definite matrix T = (t i;j ) with negative offdiagonal elements, positive row sums and satisfying certain bounding conditions that its inverse is well approximated, uniformly to order 1=n 2 ; by a matrix S = (s i;j ); where s i;j = ffi i;j =t i;i ..."
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Cited by 3 (0 self)
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It is shown for an n \Theta n symmetric positive definite matrix T = (t i;j ) with negative offdiagonal elements, positive row sums and satisfying certain bounding conditions that its inverse is well approximated, uniformly to order 1=n 2 ; by a matrix S = (s i;j ); where s i;j = ffi i;j =t i
Positive definite matrix approximation with condition number constraint POSITIVE DEFINITE MATRIX APPROXIMATION WITH CONDITION NUMBER CONSTRAINT
"... Abstract. Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one in this note when w ..."
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Abstract. Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one in this note when
Stieltjes moment sequences and positive definite matrix sequences
 Proc. Amer. Math. Soc
, 1998
"... Abstract. For a certain constant δ>0 (a little less than 1/4), every function f: N0 →]0, ∞ [ satisfying f(n) 2 ≤ δf(n − 1)f(n +1), n ∈ N, is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence f: N0 → R there is a positive definite matrix sequence (an) whi ..."
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Cited by 1 (0 self)
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Abstract. For a certain constant δ>0 (a little less than 1/4), every function f: N0 →]0, ∞ [ satisfying f(n) 2 ≤ δf(n − 1)f(n +1), n ∈ N, is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence f: N0 → R there is a positive definite matrix sequence (an
Positivedefinite matrix processes of finite variation
 Probab. Math. Statist
, 2007
"... Processes of finite variation, which take values in the positive semidefinite matrices and are representable as the sum of an integral with respect to time and one with respect to an extended Poisson random measure, are considered. For such processes we derive conditions for the square root (and the ..."
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Cited by 18 (5 self)
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Processes of finite variation, which take values in the positive semidefinite matrices and are representable as the sum of an integral with respect to time and one with respect to an extended Poisson random measure, are considered. For such processes we derive conditions for the square root (and
GENERATE A POSITIVE DEFINITE MATRIX/COVARIANCE MA
, 2013
"... Description The package contains functions for generating random clusters, generating random covariance/correlation matrices,calculating a separation index (data and population version) for pairs of clusters or cluster distributions, and 1D and 2D projection plots to visualize clusters. The packag ..."
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Description The package contains functions for generating random clusters, generating random covariance/correlation matrices,calculating a separation index (data and population version) for pairs of clusters or cluster distributions, and 1D and 2D projection plots to visualize clusters. The package also contains a function to generate random clusters based on factorial designs with factors
Learning the Kernel Matrix with SemiDefinite Programming
, 2002
"... Kernelbased learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information ..."
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Cited by 775 (21 self)
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is contained in the socalled kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input spaceclassical model selection
Families of Algorithms Related to the Inversion of a Symmetric Positive Definite Matrix
"... We study the highperformance implementation of the inversion of a Symmetric Positive Definite (SPD) matrix on architectures ranging from sequential processors to Symmetric MultiProcessors to distributed memory parallel computers. This inversion is traditionally accomplished in three “sweeps”: a Cho ..."
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Cited by 7 (3 self)
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We study the highperformance implementation of the inversion of a Symmetric Positive Definite (SPD) matrix on architectures ranging from sequential processors to Symmetric MultiProcessors to distributed memory parallel computers. This inversion is traditionally accomplished in three “sweeps”: a
A parallel primaldual interiorpoint method for semidefinite programs using positive definite matrix completion
 PARALLEL COMPUTING
, 2003
"... A parallel computational method SDPARAC is presented for SDPs (semidefinite programs). It combines two methods SDPARA and SDPAC proposed by the authors who developed a software package SDPA. SDPARA is a parallel implementation of SDPA and it features parallel computation of the elements of the Sc ..."
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Cited by 19 (12 self)
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point method using the positive definite matrix completion technique by Fukuda et al, and it performs effectively with SDPs with a large scale matrix variable, but not with a large number of equality constraints. SDPARAC benefits from the strong performance of each of the two methods. Furthermore, SDPARA
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