In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

Based on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where n × n real symmetric matrices M, C and K are constructed so that the quadratic pencil Q(λ) = λ2M + λC + K yields good approximations for the given k eigenpairs. We discuss the case where M is positive definite for 1 ≤ k ≤ n, and a general solution to this problem for n + 1 ≤ k ≤ 2n. The efficiency of our methods is illustrated by some numerical experiments.

In this short note, we study a relation between the tensor product of matrices and a multilinear map defined by the optimal operator. In this particular case, the linear transform (mediating morphism) hidden in the abstract definition of the general tensor product can be determined explicitly.

In this short note, we present a sharp upper bound for the perturbation of eigenvalues of a singular diagonalizable matrix given by Stanley C. Eisenstat [3].