scholarly journals The theory and applications of complex matrix scalings

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Rajesh Pereira ◽  
Joanna Boneng
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Diagonal Matrix ◽  
Complex Matrix ◽  
Positive Definite Matrices ◽  
Diagonal Scaling

AbstractWe generalize the theory of positive diagonal scalings of real positive definite matrices to complex diagonal scalings of complex positive definite matrices. A matrix A is a diagonal scaling of a positive definite matrix M if there exists an invertible complex diagonal matrix D such that A = D*MD and where every row and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and we conjecture that every n by n positive definite matrix has at most 2

The Judgement for Generalized Positive Definite Matrices in Signal Processing

2010 ◽  
Vol 121-122 ◽  
pp. 128-132 ◽  
Author(s):  
Xue Ting Liu
Keyword(s):  
Signal Processing ◽  
Coding Theory ◽  
Applied Mathematics ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Research Field ◽  
Positive Definite Matrices ◽  
Computational Mathematics ◽  
Special Matrix ◽  
Active Research

The generalized positive definite matrix is an active research field of special matrix, they have applied in computational mathematics, economics, physics, biology, applied mathematics, numerical computation, signal processing, coding theory, oil investigation in recent years, and so on. In this paper, motivated by [3], we give a simple and convenient judging methodwhich can be used to judge whether an nonnegative real matrix A is an generalized positive definite matrix or not.

scholarly journals On the cardinality of complex matrix scalings

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
George Hutchinson
Keyword(s):  
Upper Bound ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Complex Matrix ◽  
Real Matrices

AbstractWe disprove a conjecture made by Rajesh Pereira and Joanna Boneng regarding the upper bound on the number of doubly quasi-stochastic scalings of an n × n positive definite matrix. In doing so, we arrive at the true upper bound for 3 × 3 real matrices, and demonstrate that there is no such bound when n ≥ 4.

scholarly journals A Non-Iterative Method for the Difference of Means on the Lie Group of Symmetric Positive-Definite Matrices

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 255
Author(s):  
Xiaomin Duan ◽  
Xueting Ji ◽  
Huafei Sun ◽  
Hao Guo
Keyword(s):  
Lie Group ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Affine Invariant ◽  
Symmetric Positive Definite Matrix ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
The Mean ◽  
The Difference ◽  
Symmetric Positive Definite Matrices

A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is to transform a symmetric positive-definite matrix into a symmetric matrix via logarithmic maps, and then to transform the results back to the Lie group through exponential maps. Moreover, the present method does not need to compute the mean matrix and retains the usual Euclidean operations in the domain of matrix logarithms. In addition, for some randomly generated positive-definite matrices, the method is compared using experiments with that induced by the classical affine-invariant Riemannian metric. Finally, our proposed method is applied to denoise the point clouds with high density noise via the K-means clustering algorithm.

Positive Powers of Positive Positive Definite Matrices

1996 ◽  
Vol 48 (1) ◽  
pp. 196-209 ◽  
Author(s):  
Lon Rosen
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Positive Definite Matrices

AbstractLet C be an n x n positive definite matrix. If C ≥ 0 in the sense that Cij ≥ 0 and if p > n — 2, then Cp ≥ 0. This implies the following "positive minorant property" for the norms ‖A‖p = [tr(A*A)p/2]1/P. Let 2 < p ≠ 4, 6, … . Then 0 ≤ A ≤ B => ‖A‖p ≥ ‖B‖P if and only if n < p/2 + 1.

Variational principles for symplectic eigenvalues

2020 ◽  
pp. 1-7
Author(s):  
Rajendra Bhatia ◽  
Tanvi Jain
Keyword(s):  
Variational Principles ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Diagonal Matrix ◽  
Symplectic Matrix ◽  
Weyl Inequality

Abstract If A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$

scholarly journals On a Factorisation of Positive Definite Matrices

1963 ◽  
Vol 6 (3) ◽  
pp. 405-407 ◽  
Author(s):  
Kulendra N. Majindar
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Positive Definite Matrix ◽  
Positive Definite Matrices ◽  
Schur Product ◽  
Symmetric Positive Definite ◽  
Symmetric Positive Definite Matrices

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.

Bounds for Characteristic Values of Positive Definite Matrices

1972 ◽  
Vol 15 (1) ◽  
pp. 51-56 ◽  
Author(s):  
P. A. Binding ◽  
W. D. Hoskins ◽  
P. J. Ponzo
Keyword(s):  
Related Problem ◽  
Geometric Mean ◽  
General Setting ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Arithmetic Mean ◽  
Positive Real ◽  
Positive Definite Matrices ◽  
Smallest Eigenvalue ◽  
Characteristic Values

We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].

M-convolutions of products and ratios, statistical distributions and fractional calculus

Analysis ◽  
2016 ◽  
Vol 36 (1) ◽  
Author(s):  
Arakaparampil M. Mathai
Keyword(s):  
Fractional Calculus ◽  
Random Variables ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
The Real ◽  
Real Scalar ◽  
Scalar Variable ◽  
The Matrix

AbstractIt is shown that Mellin convolutions of products and ratios in the real scalar variable case can be considered as densities of products and ratios of two independently distributed real scalar positive random variables. It is also shown that these are also connected to Krätzel integrals and to the Krätzel transform in applied analysis, to reaction-rate probability integrals in astrophysics and to other related aspects when the random variables have gamma or generalized gamma densities, and to fractional calculus when one of the variables has a type-1 beta density and the other variable has an arbitrary density. Matrix-variate analogues are also discussed. In the matrix-variate case, the M-convolutions introduced by the author are shown to be directly connected to densities of products and ratios of statistically independently distributed positive definite matrix random variables in the real case and to Hermitian positive definite matrices in the complex domain. These M-convolutions reduce to Mellin convolutions in the scalar variable case.

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