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Bounds on the Bures–Wasserstein distance

A simple example shows the bound (2) cannot possibly hold. Consider $$ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad H = \begin{bmatrix} 1 & 0 \\ 0 & \varepsilon \end{bmatrix}...
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Positive definiteness of a matrix-valued function

First a remark: the usual and widely used definition for positive definiteness is what you wrote at the very end of your question $\underline{\rm and}$ the fact that the matrix $F(t)=(f(t_i,t_j)_{1\le ...
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5 votes

Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?

This Asymptote code seems to work perfectly and for any $t$ in your range the estimate uses $Cd$ operations and is a guaranteed upper bound though I am not sure whether $C$ is small enough for you (I ...
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4 votes

Question on density of certain set of matrices

It suffices to check whether $B^{-1}S=:S'$ has measure zero in $B^{-1}Q=:Q'$. We have $$Q'={\bf Sym}_n(\mathbb R),\qquad S'=\{{\bf Sym}_n(\mathbb R)|B(\Sigma+\Sigma^3)\in{\bf Sym}_n(\mathbb R)\}.$$ ...
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