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The goal of this problem is to see the reasoning for the term “diagonalizable," and the two seemingly different common definitions. Let 6 1 0 16 0 0 M 00 8 1 0 0 -5 2 (a) Find the characteristic polynomial of M. (b) Find all eigenvalues of M. (c) Find the algebraic and geometric multiplicities of each eigenvalue. (d) M is diagonalizable. Find an ordered eigenbasis, B, for M. (e) Let TR4 → R4 be given by T(x) = Mã. For each vector ʊ in the eigenbasis B from (d), find [T()]. (Notice you can do this without doing any matrix algebra, since if v is an eigenvector of M with eigenvalue λ, then T(√) = λʊ) (f) Use your work from (e) to find a matrix A so that [T(x)] = A[x]. (g) Let be the standard basis for R4, find P→ and Pe→. (h) Find a matrix A so that 23 [T()] = A[7] B by using the change of basis matrices from (g) and M.

The goal of this problem is to see the reasoning for the term “diagonalizable," and the two seemingly different common definitions. Let 6 1 0 16 0 0 M 00 8 1 0 0 -5 2 (a) Find the characteristic polynomial of M. (b) Find all eigenvalues of M. (c) Find the algebraic and geometric multiplicities of each eigenvalue. (d) M is diagonalizable. Find an ordered eigenbasis, B, for M. (e) Let TR4 → R4 be given by T(x) = Mã. For each vector ʊ in the eigenbasis B from (d), find [T()]. (Notice you can do this without doing any matrix algebra, since if v is an eigenvector of M with eigenvalue λ, then T(√) = λʊ) (f) Use your work from (e) to find a matrix A so that [T(x)] = A[x]. (g) Let be the standard basis for R4, find P→ and Pe→. (h) Find a matrix A so that 23 [T()] = A[7] B by using the change of basis matrices from (g) and M.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section: Chapter Questions
Problem 9RQ
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Please help with parts d,e,f with solution and steps!

The goal of this problem is to see the reasoning for the term “diagonalizable," and the two seemingly different common definitions. Let 6 1 0 16 0 0 M 00 8 1 0 0 -5 2 (a) Find the characteristic polynomial of M. (b) Find all eigenvalues of M. (c) Find the algebraic and geometric multiplicities of each eigenvalue. (d) M is diagonalizable. Find an ordered eigenbasis, B, for M. (e) Let TR4 → R4 be given by T(x) = Mã. For each vector ʊ in the eigenbasis B from (d), find [T()]. (Notice you can do this without doing any matrix algebra, since if v is an eigenvector of M with eigenvalue λ, then T(√) = λʊ) (f) Use your work from (e) to find a matrix A so that [T(x)] = A[x]. (g) Let be the standard basis for R4, find P→ and Pe→. (h) Find a matrix A so that 23 [T()] = A[7] B by using the change of basis matrices from (g) and M.
Transcribed Image Text:The goal of this problem is to see the reasoning for the term “diagonalizable," and the two seemingly different common definitions. Let 6 1 0 16 0 0 M 00 8 1 0 0 -5 2 (a) Find the characteristic polynomial of M. (b) Find all eigenvalues of M. (c) Find the algebraic and geometric multiplicities of each eigenvalue. (d) M is diagonalizable. Find an ordered eigenbasis, B, for M. (e) Let TR4 → R4 be given by T(x) = Mã. For each vector ʊ in the eigenbasis B from (d), find [T()]. (Notice you can do this without doing any matrix algebra, since if v is an eigenvector of M with eigenvalue λ, then T(√) = λʊ) (f) Use your work from (e) to find a matrix A so that [T(x)] = A[x]. (g) Let be the standard basis for R4, find P→ and Pe→. (h) Find a matrix A so that 23 [T()] = A[7] B by using the change of basis matrices from (g) and M.
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