(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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detailed explanation please!
Transcribed Image Text:(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3
a
1
(!)
0
a
1
0 0 a
Show that
p(Ma)
Ma:=
=
p(a) p'(a)p"(a)
(²
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.
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