(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 a 1 0 0 a 1 0 0 a Show that Ma := p(Ma) = ( p(a) p'(a)p"(a) 0 0 p(a) 0 p'(a) p(a). x 3 matrix (Hint: prove it first for monomials of the form x" by induction on n, and use the principle of linearity to prove this for all polynomials.) (b) Use this to determine the minimal polynomial of Ma.

(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 a 1 0 0 a 1 0 0 a Show that Ma := p(Ma) = ( p(a) p'(a)p"(a) 0 0 p(a) 0 p'(a) p(a). x 3 matrix (Hint: prove it first for monomials of the form x" by induction on n, and use the principle of linearity to prove this for all polynomials.) (b) Use this to determine the minimal polynomial of Ma.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 52EQ

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