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. (
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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![Let A, B € Rn×n and let R[x] be the set of all polynomials in variable x with coefficients in R.
Definition 1: For any p(x) = Σo C₁x¹ = R[x] define the “evaluation of p(x) at A" as
k
i=0
p(A)
:=
k
Σ GA
i=0
=
CkAk + Ck-1Ak-1 + ... + C₁ A + coỈn, (here Aº := In).
Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA.
Let A, B € Rn×n 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.
(](/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 € Rn×n and let R[x] be the set of all polynomials in variable x with coefficients in R.
Definition 1: For any p(x) = Σo C₁x¹ = R[x] define the “evaluation of p(x) at A" as
k
i=0
p(A)
:=
k
Σ GA
i=0
=
CkAk + Ck-1Ak-1 + ... + C₁ A + coỈn, (here Aº := In).
Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA.
Let A, B € Rn×n 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.
(
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