(a) For the parts (i)-(ii), let A, B E Rnxn be such that A and B commute. Furthermore, let p(x), q(x) E R[x]. (i) Prove that p(A) and B commute. (ii) Is it true that p(A) and q(B) commute? Justify your answer.
(a) For the parts (i)-(ii), let A, B E Rnxn be such that A and B commute. Furthermore, let p(x), q(x) E R[x]. (i) Prove that p(A) and B commute. (ii) Is it true that p(A) and q(B) commute? Justify your answer.
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 , \quad \text{(here \( A^0 := I_n \)).}
\]
**Definition 2:** Two matrices \( A, B \in \mathbb{R}^{n \times n} \) are said to *commute* if \( AB = BA \).
(a) For the parts (i)-(ii), let \( A, B \in \mathbb{R}^{n \times n} \) be such that \( A \) and \( B \) commute. Furthermore, let \( p(x), q(x) \in \mathbb{R}[x] \).
(i) Prove that \( p(A) \) and \( B \) commute.
(ii) Is it true that \( p(A) \) and \( q(B) \) commute? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2F007b24ed-7093-446c-a88e-1cfcc884068b%2Fyhd89ny_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 , \quad \text{(here \( A^0 := I_n \)).}
\]
**Definition 2:** Two matrices \( A, B \in \mathbb{R}^{n \times n} \) are said to *commute* if \( AB = BA \).
(a) For the parts (i)-(ii), let \( A, B \in \mathbb{R}^{n \times n} \) be such that \( A \) and \( B \) commute. Furthermore, let \( p(x), q(x) \in \mathbb{R}[x] \).
(i) Prove that \( p(A) \) and \( B \) commute.
(ii) Is it true that \( p(A) \) and \( q(B) \) commute? Justify your answer.
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