On properties of quadratic matrices
by
Alicja Smoktunowicz
Institute of Mathematics, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland
Coauthors: Marek Aleksiejczyk

Let A be a complex n ×n matrix. Assume that A satisfies a quadratic equation

(A-p I) (A- q I) = 0,

where I denotes the identity matrix.

The set of quadratic matrices includes the set of:

• projections (A² = A ),
• involutions (A² = I),
• nilpotents (A² = 0),
• elementary matrices (A = I - u w^*),

We find the closest normal matrix X to a quadratic matrix A in the 2-norm and the Frobenius norm. A matrix X is normal if X X^* = X^* X.

We prove some interesting inequalities for singular values of quadratic matrices using Schur's theorem (any complex matrix is unitarily similar to a triangular matrix). The singular values of A are the positive square roots of the eigenvalues of A^* A.

We discuss also the problem how to find a quadratic matrix with prescribed singular values.

Date received: January 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-54.