# Factorizing a Matrix Quadratic Polynomial

### New Zealand Journal of Mathematics

Vol. 43, (2013), Pages 1-6

Christopher S. Withers

Applied Mathematics Group

Industrial Research Limited

Lower Hutt,

New Zealand

School of Mathematics

University of Manchester

Manchester M13 9PL,

United Kingdom

Abstract Let A1, A2 lie in ${\mathbb{C}}^{r\times r}$ and t in ${\mathbb{C}}$. For the matrix quadratic polynomial, $I+A_1t+A_2t^2 = \left( I - \alpha_1t \right) \left(I - \alpha_2t \right),$ we give explicit solutions for 12) in ${\mathbb{C}}^{r\times r}$. We consider two cases: i) A1 and A2 commute; ii) A1 and A2 do not commute.

Keywords Factorization; Jordan blocks; Matrix quadratic equation.

Classification (MSC2000) 15B99.