# Questions tagged [hermitian-matrices]

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.

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### Eigenvalue problem and its complex conjugate

The following problem comes from a classical-mechanics treatment of lattice vibrations. It's essentially a linear algebra question, though, so I thought it would be on-topic here. If it may be assumed ...
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### Let $A$ be a Hermitian matrix, prove that $v^*Av >0$ for all $0 \not = v \in \Bbb C^n$ $\iff$ all the eigenvalues of $A$ are greater than zero

Let $A \in \Bbb C^{n \times n}$ be a Hermitian matrix, prove that $v^*Av >0$ for all $0 \not = v \in \Bbb C^n$ $\iff$ all the eigenvalues of $A$ are greater than zero I started this by stating ...
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Suppose I have two commuting Hermitian matrices $A$ and $B$: $[A,B] = 0$. I can always find a unitary operator $U$ such that simultaneously diagonalize both matrices, i.e., U^* A U = ...
### If $A$ is Hermitian and $U=(A-iI)(A+iI)^{−1}$ is unitary, then $U-I$ is invertible
I'm stuck on the $3$rd part of a question, the first part was proving $A-iI$ is invertible when $A$ is Hermitian, second was proving $U=(A-iI)(A+iI)^{−1}$ is unitary. Now I'm being asked to prove $U-I$...