Maximum permanent and Hermitian matrices

We study the reduction from a bipartite Hermitian matrix H to two single-qubit reduced Hermitian matrices HA and HB and the inverse problem. The inverse problem extends the so-called marginal problem from quantum information theory. We explicitly obtain the expressions of the matrices mentioned above and then extend them to positive semidefinite matrices by showing several inequalities involving a few parameters from the matrices. We also extend our results to multi-quit matrices. Our study constructs the intimate connection between the generalized marginal problems in terms of Hermitian matrices and the traditional marginal problem in terms of positive semidefinite matrices.

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.

Given a partition a = {ax , a2, ... , as} , ax > a2 > > as, of n we let X denote the derived irreducible character of S" , and we associate with a a derived partition a = {a, -1, a2 -1, .. , ott -1 , at+x , ... , as , 1 } where t denotes the smallest positive integer such that at > a(+1 (aJ+r = 0).We show that if y is a decomposable C-valued n-linear function on Cm x Cm x x Cm (-copies) then (Xa Y, Y) > (Xa, Y, Y) .Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions dx and dx ( , namely that (Xa(e))-xdx(B)>(Xa,(e))-ldxAB) for each n x n positive semidefinite Hermitian matrix B .This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.

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.

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

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

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.

Williamson's theorem states that every real symmetric positive definite matrix $A$ of even order can be brought to diagonal form via a symplectic $T$-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of $A$. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.

Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.

Norms and inequalities for condition numbers, III

Sparse matrix decompositions and graph characterizations

It is studied to find a positive semidefinite(not necessarily symmetric) matrix A satisying equation AX=B for given X, B. The solvable conditions are given and a general expression of the solutions is provided. The problem to find the positive semidefinite matrix  which minimizes A∗-A in Frobenius norm is considered, where matrix A satisfies AX=B. Some numerical results are also reported.

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.

Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that given merely the desirable structure of the spectrum, namely given the cardinalities of real or complex eigenvalues but not of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed.

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.