The estimation of the positive definite solutions to perturbed discrete Lyapunov equations is discussed. Several upper bounds of the positive definite solutions are obtained when the perturbation parameters are norm-bounded uncertain. In the derivation of the bounds, one only needs to deal with eigenvalues of matrices and linear matrix inequalities, and thus avoids solving high-order algebraic ...

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. In this study the upper bounds for the spectral radius of weighted graphs, which edge weights are positive definite matrices, are compared. Mathematics Subject Classification: 05C50

Two-stage iterative methods for the solution of linear systems are studied. Convergence of both stationary and nonstationary cases is analyzed when the coefficient matrix is Hermitian positive definite. Keywords—Linear systems, Hermitian matrices, Positive definite matrices, Iterative methods, Nonstationary methods, Two-stage methods.

In this paper sufficient conditions are derived for the existence of unique and positive definite solutions of the matrix equations X−A1XA1− . . .−A∗mXAm = Q and X+A1XA1+ . . .+ A∗mXAm = Q. In the case there is a unique solution which is positive definite an explicit expression for this solution is given.

The Line Sum Scaling problem for a nonnegative matrix A is to find positive definite diagonal matrices Y , Z which result in prescribed row and column sums of the scaled matrix Y AZ. The Matrix Balancing problem for a nonnegative square matrix A is to find a positive definite diagonal matrix X such that the row sums in the scaled matrix XAX are equal to the corresponding column sums. We demonst...

In this paper sufficient conditions are derived for the existence of unique and positive definite solutions of the matrix equations X−A1XA1− . . .−A∗mXAm = Q and X+A1XA1+ . . .+ A∗mXAm = Q. In the case there is a unique solution which is positive definite an explicit expression for this solution is given.

An analytic proof is proposed of Wiener’s theorem on factorization of positive definite matrix-functions.

Notation: The set of real symmetric n ×n matrices is denoted S . A matrix A ∈ S is called positive semidefinite if x Ax ≥ 0 for all x ∈ R, and is called positive definite if x Ax > 0 for all nonzero x ∈ R . The set of positive semidefinite matrices is denoted S and the set of positive definite matrices + n is denoted by S++. The cone S is a proper cone (i.e., closed, convex, pointed, and solid). +

A (semistability) factor [semifactor] of a matrix AE r: nn is a positive definite [positive semidefinite] matrix H such that AH + HA* is positive semidefinite. We give three proofs to show that if A has a semistability factor then it cannot be unique. We give necessary and sufficient conditions for a matrix H to be a (semi)factor of a given matrix. We also determine the dimension of the cone of...