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. (
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 65E
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