Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real.

djeljenike 2021-03-06 Answered
Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real.
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Expert Answer

oppturf
Answered 2021-03-07 Author has 94 answers
Step 1
The given statement is False
Step 2
Counter Example:
Let A=[22i2i0] and B=[22i2i0]
A and B are Hermitian matrices
Now consider the product AB
AB=[22i2i0][22i2i0]=[4+4i24i04i04i2]
AB=[444i4i4]=[04i4i4]
Consider characteristic equation for the matrix AB to get eigenvalue
det(AλI)=|0λ4i4i4λ|=|λ4i4i4λ|=0
λ(4λ)(16i2)=0
4λ+λ2+16=0
λ2+4λ+16=0
λ=4±424(16)2
λ=4±16642
λ=4±48i2=2±23i
Clearly the eigenvalues are not real numbers.
Step 3
Answer:
The given statement is False.
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Jeffrey Jordon
Answered 2022-01-24 Author has 2064 answers

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