If A is diagonalizable and for all eigenvalues , lambda text{ of } A , |lambda|=1 , then A is unitary. True or False?

BenoguigoliB 2021-01-04 Answered
If A is diagonalizable and for all eigenvalues , λ of A,|λ|=1 , then A is unitary. True or False?
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Expert Answer

FieniChoonin
Answered 2021-01-05 Author has 102 answers
Step 1
Solution.
Given: A is diagonalizable and all eigenvalue λ of A,|λ|=1
Step 2
Let A=[1101] , the eigenvalue of A is 1,-1 we know that if A has distinct eigenvalue then A is diagonalizable.
A is diagonalizable , and 1,-1 is eigenvalue of A also |1|=1,|1|=1
Now check A is unitary?
If A is unitary then AAT=I ,
AAT=[1101][1101]T=[1101][1011]=[2111][1001]
A is not unitary
Hence if A is diagonalizable and all eigenvalue
λ of A,|λ|=1
then A need not be unitary
The given statement is False
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Jeffrey Jordon
Answered 2022-01-23 Author has 2087 answers

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