The polar decomposition of block companion matrices Gregory Kalogeropoulos1 and Panayiotis Psarrakos2 September 28, 2005 Abstract m m¡1 Let L(¸) = In¸ + Am¡1¸ + ¢ ¢ ¢ + A1¸ + A0 be an n £ n monic matrix polynomial, and let CL be the corresponding block companion matrix. In this note, we extend a known result on scalar polynomials to obtain a formula for the polar decomposition of CL when the matrices A0 Pm¡1 ¤ and j=1 Aj Aj are nonsingular. Keywords: block companion matrix, matrix polynomial, eigenvalue, singular value, polar decomposition. AMS Subject Classifications: 15A18, 15A23, 65F30. 1 Introduction and notation Consider the monic matrix polynomial m m¡1 L(¸) = In¸ + Am¡1¸ + ¢ ¢ ¢ + A1¸ + A0; (1) n£n where Aj 2 C (j = 0; 1; : : : ; m ¡ 1; m ¸ 2), ¸ is a complex variable and In denotes the n £ n identity matrix. The study of matrix polynomials, especially with regard to their spectral analysis, has a long history and plays an important role in systems theory [1, 2, 3, 4]. A scalar ¸0 2 C is said to be an eigenvalue of n L(¸) if the system L(¸0)x = 0 has a nonzero solution x0 2 C . This solution x0 is known as an eigenvector of L(¸) corresponding to ¸0. The set of all eigenvalues of L(¸) is the spectrum of L(¸), namely, sp(L) = f¸ 2 C : detL(¸) = 0g; and contains no more than nm distinct (finite) elements. 1Department of Mathematics, University of Athens, Panepistimioupolis 15784, Athens, Greece (E-mail:
[email protected]). 2Department of Mathematics, National Technical University, Zografou Campus 15780, Athens, Greece (E-mail:
[email protected]).