For a polynomial p(x) = anx" + an-1xn-1 +... + a₁x + ao with a; E F and square matrix A with entries in F, define p(A) by p(A) = an A+an-1A-1 + + a₁A+aoI. Let A € Mnxn(C) be diagonalizable and c(x) the characteristic polynomial of A. Then show (ie, prove) that c(A) = 0, where O is the n x n zero matrix.
For a polynomial p(x) = anx" + an-1xn-1 +... + a₁x + ao with a; E F and square matrix A with entries in F, define p(A) by p(A) = an A+an-1A-1 + + a₁A+aoI. Let A € Mnxn(C) be diagonalizable and c(x) the characteristic polynomial of A. Then show (ie, prove) that c(A) = 0, where O is the n x n zero matrix.
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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![For a polynomial \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with \( a_i \in \mathbb{F} \) and square matrix \( A \) with entries in \( \mathbb{F} \), define \( p(A) \) by
\[
p(A) = a_nA^n + a_{n-1}A^{n-1} + \cdots + a_1A + a_0I.
\]
Let \( A \in M_{n \times n}(\mathbb{C}) \) be diagonalizable and \( c(x) \) the characteristic polynomial of \( A \). Then show (i.e., prove) that \( c(A) = O \), where \( O \) is the \( n \times n \) zero matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2F7b6c656b-4636-4f4e-ab03-ed26c99b4746%2F4pnozoo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For a polynomial \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with \( a_i \in \mathbb{F} \) and square matrix \( A \) with entries in \( \mathbb{F} \), define \( p(A) \) by
\[
p(A) = a_nA^n + a_{n-1}A^{n-1} + \cdots + a_1A + a_0I.
\]
Let \( A \in M_{n \times n}(\mathbb{C}) \) be diagonalizable and \( c(x) \) the characteristic polynomial of \( A \). Then show (i.e., prove) that \( c(A) = O \), where \( O \) is the \( n \times n \) zero matrix.
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