Linear Algebra - Hermitian matrices

[big man]

Sorry if this is in the wrong section. I just want to check my answer since I've been going through the exam.


Given that A is an n × n matrix and I is the n × n identity matrix, select all the correct responses below.
A Every diagonalisable matrix is normal.
B If A is Hermitian, then $$A^TA$$ is also Hermitian.
C If all eigenvalues of A have algebraic multiplicity 1, then it is diagonalisable.
D If A is Hermitian, then A + I is always invertible.
E If B = A*A, then B is Hermitian.
F None of the above

I know that A is wrong because it is only every unitarily diagonalisable matrix that is normal. I think B is correct because if it is Hermitian it means that A is equal to the complex conjugate of the transpose. So really A is a symmetric matrix and $$A^T=A=A^*$$. Not too sure about C in the end. I think D is wrong because the Hermitian matrix doesn't have to be invertible to begin with. E is also wrong because you don't know enough information about the matrix B to make that statement. Obviously the selection then can't be "none of the above".

I think I might have them right, but I would like to check. ALthough I don't actually know about C so if someone could explain that to me I'd be grateful.

Thanks

 

[Gokul43201]

Sorry if this is in the wrong section. I just want to check my answer since I've been going through the exam.


Given that A is an n × n matrix and I is the n × n identity matrix, select all the correct responses below.

B If A is Hermitian, then $$A^TA$$ is also Hermitian.

I think B is correct because if it is Hermitian it means that A is equal to the complex conjugate of the transpose. So really A is a symmetric matrix and $$A^T=A=A^*$$.

No, A need not be symmetric. Where did you get that from ? It doesn't say anywhere that A is real.

PS : You are correct that choice A (all diag matrices are normal) is wrong.

 

[big man]

Oh I just thought that any matrix where $$A^T=A$$ is called a symmetric matrix.

 

[Gokul43201]

Oh I just thought that any matrix where $$A^T=A$$ is called a symmetric matrix.

That's right, but where is that given in the question ?

 

[big man]

Ahh wait yeah I'm an idiot...I just took something out of the statement that wasn't there. It doesn't equal just A transpose. It's only the complex conjugate of the transpose. OK so B is actually wrong as well.

 

[Hurkyl]

For the ones you think are wrong, have you tried coming up with a counterexample?

 

[big man]

haha thanks for that. I just tried some examples for E and found that it was true. Also found a proof in the notes that C was true so that was good. So yeah I found that C and E are the actual answers (I'm pretty damn sure about this)...can't believe how off my answers were at the beginning.

Thanks again