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) = ₁0 Cx² € R[x] define the "evaluation of p(x) at A” as i=0 p(A) = k Σ GA i=0 = CkAk + CK-1 + +₁A+ coIn, (here Aº = In). = Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA. Let A Rnxn. Prove that there exists a polynomial p(x) = R[x] of degree at most n²+1 such that p(A) = Onxn. Hint: Consider {In, A, A², An², An²+1} and use the fact that dim(Rn×n) = n². (
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) = ₁0 Cx² € R[x] define the "evaluation of p(x) at A” as i=0 p(A) = k Σ GA i=0 = CkAk + CK-1 + +₁A+ coIn, (here Aº = In). = Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA. Let A Rnxn. Prove that there exists a polynomial p(x) = R[x] of degree at most n²+1 such that p(A) = Onxn. Hint: Consider {In, A, A², An², An²+1} and use the fact that dim(Rn×n) = n². (
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 \( A \in \mathbb{R}^{n \times n} \).
*Prove that there exists a polynomial \( p(x) \in \mathbb{R}[x] \) of degree at most \( n^2 + 1 \) such that \( p(A) = 0_{n \times n} \).*
**Hint:** Consider \( \{I_n, A, A^2, \ldots, A^{n^2}, A^{n^2+1} \} \) and use the fact that \( \dim(\mathbb{R}^{n \times n}) = n^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2Fb71cad82-18c5-4d26-889b-b06629437b11%2Fh207jl_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 \( A \in \mathbb{R}^{n \times n} \).
*Prove that there exists a polynomial \( p(x) \in \mathbb{R}[x] \) of degree at most \( n^2 + 1 \) such that \( p(A) = 0_{n \times n} \).*
**Hint:** Consider \( \{I_n, A, A^2, \ldots, A^{n^2}, A^{n^2+1} \} \) and use the fact that \( \dim(\mathbb{R}^{n \times n}) = n^2 \).
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