Answered: For each of the following matrices A,…

For each of the following matrices A, find an orthogonal or unitary matrix P and a diagonal matrix D such that P* AP D. (1 2) 3 — 3і (b) (c) (a) 2 %3D 2. 3 + 3i (d) (e) 1 2 0 2 1 2 1 2 2 0/ 2 ||

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.2: Diagonalization

Problem 39E: Are the two matrices similar? If so, find a matrix P such that B=P1AP. A=[100020003]B=[300020001]

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Question

For each of the following matrices A, find an orthogonal or unitary matrix P
and a diagonal matrix D such that P* AP
D.
(1 2)
3 — 3і
(b)
(c)
(a)
2
%3D
2.
3 + 3i
(d)
(e)
1
2 0 2
1 2 1
2 2 0/
2
||

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Let A,D, and P be nn matrices satisfying AP=PD. Assume that P is nonsingular and solve this for A. Must it be true that A=D?
a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.
Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.
Are the two matrices similar? If so, find a matrix P such that B=P1AP. A=[100020003]B=[300020001]