COMPLETENESS THEOREM FOR A PRODUCT OF SELF-ADJOINT MATRICES

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

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.

Modeling of multicomponent batch distillation processes with chemical reaction and their control systems

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

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.

Simulation of multicomponent batch distillation processes

Norms and inequalities for condition numbers, III

Sparse matrix decompositions and graph characterizations

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.

Maximum permanent and Hermitian matrices

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.

The goal of this chapter is to show that there are nice normal forms for symmetric matrices, skew-symmetric matrices, orthogonal matrices, and normal matrices. The spectral theorem for symmetric matrices states that symmetric matrices have real eigenvalues and that they can be diagonalized over an orthonormal basis. The spectral theorem for Hermitian matrices states that Hermitian matrices also have real eigenvalues and that they can be diagonalized over a complex orthonormal basis. Normal matrices can be block diagonalized over an orthonormal basis with blocks having size at most two, and there are refinements of this normal form for skewsymmetric and orthogonal matrices.

An inertia theorem for Lyapunov's equation and the dimension of a controllability space

In this article, we obtain another proof of the following classical trace inequality which says that if A is a positive semidefinite matrix and B is a Hermitian matrix, then tr A? BA?B ? tr A? BA?B for all non-negative real numbers ?, ?, ?, ? for which ? + ? = ? + ? and max {?,?} ? max {?,?}.This is a generalization of trace inequalities due to T. Ando, F.Hiai, and K. Okubo for the special cases when ?= ?+?,?=0 and when ?= ?=?+?/2, namely tr (A?+?/2B)2 ? tr A?BA?B? tr A?+?B2.

In this study, we consider a non-linear matrix equation of the formwhere Q is a Hermitian positive definite matrix, A * i stands for the conjugate transpose of an n n matrix A i and F j are order-preserving continuous mappings from the set of all Hermitian matrices to the set of all positive definite matrices such that F (O) = O.We discuss sufficient conditions that ensure the existence of a unique positive definite solution of the given matrix equation.For this, we derive some fixed point results for Suzuki-implicit type mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order).We provide adequate examples to validate the fixed-point results and the importance of related work, and the convergence analysis of non-linear matrix equations.