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If the matrices are real and the function you have in mind is real-valued, then you indeed get the characterization you suggested. This was first shown in "I. J. Schoenberg. Positive definite function …

$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ …

The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2, …

Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the squ …

This answer uses a lot of the same ideas as Onur Oktay's answer, but I believe corrects some problems with it. If we define $\mathbf{e}_1 = (1,0)$, $\mathbf{e}_2 = (0,1)$, $\mathbf{u} = (1,1)/\sqrt{2} …

It turns out that this norm can be computed efficiently (i.e., it is in $P$). This wasn't known at the time that the Davidson paper (originally linked in a comment above) was written, which is why it …

Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$: $$ f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w …

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary o …

$\mathcal{C}$ is simply the span of the two maps that you noted (the identity and the trace) -- there is nothing else in the commutant. One (admittedly somewhat roundabout) way of seeing this is to n …

No, this is not true even if the matrices are $2 \times 2$ and $\alpha = 1/2$. For a concrete counter-example, consider $$A = \begin{bmatrix}1 & -\sqrt{3} \\ -\sqrt{3} & 3\end{bmatrix}, B = \begin{bma …