Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

The triangle graph $T_6$ is not SPN

A real symmetric matrix $A$ is copositive if $x'Ax \geq 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative matrix. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than $4$. A graph $G$ is an SPN graph if every copositive matrix whose graph is $G$ is SPN. We show that the triangle graph $T_6$ is not SPN.

A trace bound for integer-diagonal positive semidefinite matrices

We prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.