Complementary basic matrices*1

Complementary basic matrices

We show that an n× n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions (except the first and last) completely between the subdiagonal and superdiagonal part, then this matrix can be factorized into a product of n− 1 matrices, each consisting of a 2 × 2 principal submatrix in two consecutive rows (and columns) in all possible of the...

Open Sets of Conservative Matrices1

In this paper we present two principal results. In Theorem 1 we show that if r¡ denotes the open set of (conservative) matrices which map some subspace of c of infinite deficiency isomorphically onto c and if A denotes the closed set of matrices which sum some bounded divergent sequence, then [r']~=A. In Theorem 2 we produce a class of triangular matrices with the property that no triangular ma...

On Commutators and Jacobi Matrices1

The closure, W, oí the set of values (Cx, x) when ||x|| = 1 is a closed convex set (Hausdorff, cf. [8, p. 34]). A complex number z will be said to belong to the interior of W if z is in W and if one of the following conditions holds: (i) If W is two-dimensional, then z does not lie on the boundary of W; (ii) If IF is a line segment, then z is not an end point; (iii) W consists of z alone. It wa...

Hausdorff-analytic Functions of Matrices1

D-Stability of Polynomial Matrices1

Necessary and suucient conditions are formulated for the zeros of an arbitrary polynomial matrix to belong to a given region D of the complex plane. The conditions stem from a general optimization methodology mixing LFRs, rank-one LMIs and the S-procedure. They are expressed as an LMI feasibility problem that can be tackled with widespread powerful interior-point methods. Most importantly, the ...