Prove that Hermitian/Skew Herm/Unitary Matrix is a Normal Matrix

In summary, to prove that a matrix is normal, it must satisfy the condition A*A=AA*. To show that it is Hermitian, A must be equal to its conjugate transpose A*. For a matrix to be skew Hermitian, A must be equal to the negative of its conjugate transpose -A*. A matrix is unitary if it is both Hermitian and skew Hermitian. It is important to consider the basic matrix itself in the proof, rather than getting caught up in more complex calculations.
  • #1
Wildcat04
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Homework Statement


Show the proof that the following are all Normal Matrices
a. Hermitian
b. Skew Hermitian
c. Unitary

Homework Equations



Normal Matrices: A*A=AA*
Hermitian Matrices: A=A* or aij=a*ji
Skew Hermitian Matrices A=-A* or aij=-a*ji

The Attempt at a Solution



So far I have tried using the above information for Hermitian Matrices to try and prove that A*A=AA* but I keep getting answers I know not to be correct. I would appreciate a nudge in the correct direction so I can quit pulling my hair out. I have a feeling when I find the correct proof it will be quite obvious, but right now I am just missing something.
 
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  • #2
If A is hermitian then A*=A. A*A=AA=AA*. How are you trying to do it?
 
  • #3
I was trying to do it much more in depth using amn. It never occurred to me to use the basic matrix itself. Like I said, it was right in front of me. Thank you for the help!
 

What is a Hermitian matrix?

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. In other words, if A is a Hermitian matrix, then A* (conjugate transpose of A) = A.

What is a Skew Hermitian matrix?

A Skew Hermitian matrix is a complex square matrix that is equal to the negative of its own conjugate transpose. In other words, if A is a Skew Hermitian matrix, then A* = -A.

What is a Unitary matrix?

A Unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. In other words, if A is a Unitary matrix, then A*A = A*A = I (identity matrix).

What does it mean for a matrix to be normal?

A normal matrix is a square matrix that commutes with its conjugate transpose. In other words, if A is a normal matrix, then A*A = A*A*. This means that the matrix and its conjugate transpose can be written in any order without changing the result.

Why is it important for a Hermitian/Skew Hermitian/Unitary matrix to be normal?

It is important for a Hermitian/Skew Hermitian/Unitary matrix to be normal because it allows for easier computation and manipulation. Normal matrices have many useful properties, such as orthogonality and spectral decomposition, which make them valuable in various applications in mathematics and science.

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