Partitions, irreducible characters, and inequalities for generalized matrix functions

Maximum permanent and Hermitian matrices

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number , positive semi-definite matrices , , and generalized matrix functions such as the determinant and permanent, etc., we have A general scheme is introduced to prove more general inequalities involving m positive semi-definite matrices for that extend the results of other authors.

Fractional‐order circuits find a widespread use in different engineering applications. The problem of realising fractional‐order circuits has been discussed by several authors, however, it is far from being solved. Realising fractional‐order resistorless passive network with three element orders is been studied. At first, this study extends the two‐variable reactance matrix synthesis method to three‐variable case, and then proposes a synthesis method of fractional‐order reactance matrix with three element orders by variable substitution. The process in above methods mainly involves variable substitution, decomposition of three‐variable reactance matrix, extraction of unit inductors, Laurent series expansion, spectral factorisation of two‐variable positive semidefinite Hermitian matrix and synthesis of univariable reactance matrix. Then the above‐mentioned synthesis process is illustrated by two examples.

A partial ordering of CS m defined by the generalized matrix functions

Inequalities and identities for generalized matrix functions

Inequalities which compare elements of the convex cone of positive semidefinite hermitian matrices with products of roots of elements are proved. They yield inequalities for Schur functions (generalized matrix functions) which, when specialized to the determinant, give a result of R. Bellman and L. Mirsky.

For normalised generalized matrix functions f and g, we say that f dominates g if f(A)⩾g(A) for every M-matrix A. We first demonstrate a finite set of test matrices for any such inequality. Then, using results from group representation theory, all comparisons among immanants in certain classes are determined. This work parallels ongoing research into gmf inequalities on positive semidefinite matrices, for which no finite set of test matrices is available. However, the inequalities for the two classes are quite different, and the test matrices permit more rapid progress in the M-matrix case. Just as in the positive semidefinite case, the gmf inequalities we prove may be used to verify previously unknown determinantal inequalities for M-matrices, such as the symmetrized Fischer inequalities recently proved in the positive semidefinite case.

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n × n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

Sparse matrix decompositions and graph characterizations

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.

A few basic results regarding the ranks of certain types of matrices are presented. Also, some bounds on the ranks of the real and imaginary parts of a positive semi-definite Hermitian matrix are worked out. Applications of particular cases of these results in the areas like passive network synthesis, generation of two-variable reactance matrices, etc., are pointed out.

In this paper, we propose a disturbance covariance matrix estimation method for radar signal processing when the number of training samples is severely limited. Several maximum likelihood (ML) estimation methods based on the structured of the covariance matrix, which is modeled as a sum of an unknown positive semi-definite Hermitian matrix and a matrix proportional to identity, have been proposed to solve the problem. Based on the covariance structure, we incorporate additional constraint on the low rank property of the covariance matrix, and propose a low rank regularized ML estimation method for covariance matrix estimation. The low rank is achieved by solving a reweighted trace minimization problem. Experimental results demonstrate the effective of the proposed method, both in the accuracy of rank estimation and the improvement of signal interference and noise ratio (SINR).

If A is an n-square positive semidefinite Hermitian matrix of rank 1, then the Hadamard square root of A is the n-square matrix obtained by replacing each entry of A by the principal value of its square root. It is proved that if A has no zero or negative entries, then the Hadamard square root has odd rank and all odd ranks are possible.

In this paper, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(N-K)(2K+1) + K^{2}$</tex-math></inline-formula> measurements to guarantee exact recovery of the PSD matrix, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.