Let A, B E Rnxn and let R[x] be the set of all polynomials in variable x with coefficients in R. Definition 1: For any p(x) = 0 C₂x² € R[x] define the "evaluation of p(x) at A" as vi=0 p(A) := k Σ c₁A¹ = c₂A¹ + Ck−1A² i=0 ++C₁A+ coIn, (here Aº = In). = Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA. Let A, B € Rnxn be similar. Show that for any polynomial p(x) = R[x] that p(A) and p(B) are similar. Specifically, show that if B = Q-¹AQ, then p(B) = Q¯¹p(A)Q. (
Let A, B E Rnxn and let R[x] be the set of all polynomials in variable x with coefficients in R. Definition 1: For any p(x) = 0 C₂x² € R[x] define the "evaluation of p(x) at A" as vi=0 p(A) := k Σ c₁A¹ = c₂A¹ + Ck−1A² i=0 ++C₁A+ coIn, (here Aº = In). = Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA. Let A, B € Rnxn be similar. Show that for any polynomial p(x) = R[x] that p(A) and p(B) are similar. Specifically, show that if B = Q-¹AQ, then p(B) = Q¯¹p(A)Q. (
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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![Let \( A, B \in \mathbb{R}^{n \times n} \) and let \( \mathbb{R}[x] \) be the set of all polynomials in variable \( x \) with coefficients in \( \mathbb{R} \).
**Definition 1:** For any \( p(x) = \sum_{i=0}^{k} c_i x^i \in \mathbb{R}[x] \), define the "evaluation of \( p(x) \) at \( A \)" as
\[
p(A) := \sum_{i=0}^{k} c_i A^i = c_k A^k + c_{k-1} A^{k-1} + \cdots + c_1 A + c_0 I_n,
\]
(here \( A^0 := I_n \)).
**Definition 2:** Two matrices \( A, B \in \mathbb{R}^{n \times n} \) are said to commute if \( AB = BA \).
Let \( A, B \in \mathbb{R}^{n \times n} \) be similar. Show that for any polynomial \( p(x) \in \mathbb{R}[x] \) that \( p(A) \) and \( p(B) \) are similar. Specifically, show that if \( B = Q^{-1} A Q \), then \( p(B) = Q^{-1} p(A) Q \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2Fcb5c3ac9-870f-4788-9ddd-7cc0a7a390a7%2Fnfrh4to_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( A, B \in \mathbb{R}^{n \times n} \) and let \( \mathbb{R}[x] \) be the set of all polynomials in variable \( x \) with coefficients in \( \mathbb{R} \).
**Definition 1:** For any \( p(x) = \sum_{i=0}^{k} c_i x^i \in \mathbb{R}[x] \), define the "evaluation of \( p(x) \) at \( A \)" as
\[
p(A) := \sum_{i=0}^{k} c_i A^i = c_k A^k + c_{k-1} A^{k-1} + \cdots + c_1 A + c_0 I_n,
\]
(here \( A^0 := I_n \)).
**Definition 2:** Two matrices \( A, B \in \mathbb{R}^{n \times n} \) are said to commute if \( AB = BA \).
Let \( A, B \in \mathbb{R}^{n \times n} \) be similar. Show that for any polynomial \( p(x) \in \mathbb{R}[x] \) that \( p(A) \) and \( p(B) \) are similar. Specifically, show that if \( B = Q^{-1} A Q \), then \( p(B) = Q^{-1} p(A) Q \).
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