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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
0
votes
1
answer
106
views
Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w …
4
votes
Accepted
Monotonicity of matrix conjugation
No, this is not true even if the matrices are $2 \times 2$ and $\alpha = 1/2$. For a concrete counter-example, consider
$$A = \begin{bmatrix}1 & -\sqrt{3} \\ -\sqrt{3} & 3\end{bmatrix}, B = \begin{bma …
5
votes
Dual norm of a subspace of $\ell_\infty^3$
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} …
3
votes
0
answers
252
views
Schur-Horn theorem for principal submatrices
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, …
4
votes
0
answers
107
views
Characterization of "PSD-Squared" Matrices
$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$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 $n \times n$ …
5
votes
Commutant of the conjugations by unitary matrices
$\mathcal{C}$ 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 …
51
votes
2
answers
5k
views
A strengthening of the Cauchy-Schwarz inequality
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 …
8
votes
Accepted
elementwise functions of positive definite matrix
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 …
14
votes
How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?
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 …
5
votes
0
answers
147
views
Groups of operators between local unitaries and full unitaries
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 …