We derive the matrix that minimizes a quadratic criterion subject to linear equality constraints. Since we do not assume that the matrix of the quadratic form criterion is nonsingular, our optimization result is more general than those previously published. We make use of the extended result of this paper to (i) obtain the best linear unbiased estimate of the parameters in a linear regression w...

Abstract. If an n× n complex matrix A is nonsingular, then for every integer m > 1, A has an mth root B, i.e., Bm = A. In this paper, we present a new simple proof for the Jordan canonical form of the root B. Moreover, a necessary and sufficient condition for the existence of mth roots of a singular complex matrix A is obtained. This condition is in terms of the dimensions of the null spaces of...

Polar decompositions X = U A of real and complex matrices X with respect to the scalar product generated by a given indefinite nonsingular matrix Hare studied in the following special cases: (1) X is an H-contraction, (2) X is an H-plus matrix, (3) H has only one positive eigenvalue, and (4) U belongs to the connected component of the identity in the group of H-unitary matrices. Applications to...

In this paper we extend work of Kawamura, see [3], for CuntzKrieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infini...

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system – open-closed string theory. Nonperturbative solutions of the multi-critical unitary matrix models map to nonsingular solutions of the ‘renormalisation group’ equation for the string susceptibility, [P̃ , Q] =Q . We also demonstrate that the large N solutions of unitary matr...

In this paper, state-derivative and especially output-derivative feedbacks for linear time-invariant systems are derived using control approach similar to linear quadratic regulator (LQR). The optimal feedback gain matrices are derived for the desired performance. This problem is always solvable for any controllable system if the open-loop system matrix is nonsingular. Explicit expression of th...

In this note we prove NP-hardness of the following problem: Given a set of matrices, is there a convex combination of those that is a nonsingular M-matrix? Via known characterizations of M-matrices, our result establishes NP-hardness of several fundamental problems in systems analysis and control, such as testing the instability of an uncertain dynamical system, and minimizing the spectral radi...

Given a square, nonsingular matrix of univariate polynomials F ∈ K[x]n×n over a field K, we give a deterministic algorithm for finding the determinant of F. The complexity of the algorithm is O ̃(nωs) field operations where s is the average column degree or the average row degree of F. Here O ̃ notation is Big-O with log factors omitted and ω is the exponent of matrix multiplication.

This paper presents a method based on elementary transformations which may be applied to a matrix A, whose characteristic polynomial has been decomposed into linear factors, in order to obtain a nonsingular matrix P such that P-1AP is in Jordan normal form. This method can be used in the classroom, among other problems, to directly solve a linear ODE with constant coefficients. We also present ...