Suppose that A and B are diagonalizable matrices. Prove or disprove that A is similar to B if and only if A and B are unitarily equivalent.

Efan Halliday 2021-01-02 Answered
Suppose that A and B are diagonalizable matrices. Prove or disprove that A is similar to B if and only if A and B are unitarily equivalent.
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SchulzD
Answered 2021-01-03 Author has 83 answers
Step 1
Consider A and B are matrices that are diagonalizable.
The objective is to prove or disprove that A is similar to B only if A and B are unitarily equivalent.
Consider the two diagonalizable matrices are,
A=[1100]
B=[1000]
Consider that there is an invertible matrix,
P=[1101] Step 2
The matrix A is similar to B as shown below,
[1000]=[1101]1[1101][1101]
=[1101][1000]
=[1000]
The A and B are similar matrices.
But A and B are not unitary because B is symmetric, but A is not.
Also, recall the result, If B and A are unitarily equivalent.
i,j=1n|bij|2=i,j=1n|aij|2
Since, 2 is not equal then,
i,j=1n|bij|2i,j=1n|aij|2
Since,
A and B are not unitarily equivalent.
Therefore, it is not necessary that A is similar to B if and only if A and B are unitarily equivalent.
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Jeffrey Jordon
Answered 2022-01-23 Author has 2027 answers

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