# Questions tagged [hermitian-matrices]

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

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### Inequality for Hermitian matrices

Does the following equality hold for $A,B$ ( both Hermitian matrices): $$\vert \text{tr}(AB) \vert \leq \displaystyle\left(\text{tr}(\vert A\vert^k) \text{tr}(\vert B \vert ^k)\right)^{\frac{1}{k}}?$$ ...
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### A question on inequalities associated to hermitian positive semidefinite matrices

Let $\mathcal{H}_N$ denote the cone of hermitian $N$ by $N$ positive semidefinite matrices and let $H \in \mathcal{H}_N$. We associate to $H$ the following complex numbers: \begin{align*} c_1 &= ...
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The space of $N \times N$ Hermitian matrices can be construed as a real vector space with dimension $N^2$. One can define an inner product on this real vector space by $\langle A, B \rangle := trace(A^... 1 vote 1 answer 44 views ### Minimal spanning set ("conical basis") for 2x2 Hermitian PSD (positive semi-definite) cone? A linear combination$ax + by$is called conic(al) if$a, b \ge 0$(cf. section 2.1.5 of Boyd, Vandenberghe). I.e. conic(al) combinations are just linear combinations where the coefficients are ... 0 votes 0 answers 23 views ### If two hermitian matricies commute, do they share an orthonormal eigenbasis? Heads up: I'm a physicist. I know that if$A=A^\dagger$,$B=B^\dagger$commute ($AB=BA$) then there is an eigenbasis$\{v_1,v_2,...,v_n\}$which diagonalises both$A$and$B\$ even if they are ...
For homework, I'm given a matrix $$A = \begin{bmatrix} 3 & 2i & -2i\\ -2i & 0 & -1\\ 2i & -1 & 0 \end{bmatrix}$$ in an hermitian space. We are trying to find an orthonormal ...