CONVEX AND CONCAVE FUNCTIONS OF SINGULAR
VALUES OF MATRIX SUMS
This question comes from the proof of theorem $1$ linked above.
Assume $A,B,C=A+B$ are $n\times n$ Hermitian positive semidefinite matrices with eigenvalues $\alpha _1\geqslant \alpha _2\geqslant \cdots \geqslant \alpha _n,\beta _1\geqslant \beta _2\geqslant \cdots \geqslant \beta _n,\gamma _1\geqslant \gamma _2\geqslant \cdots \geqslant \gamma _n$ respectively.
Since $A$ and $B$ are semidefinite, we may write $A = A_1A_1^{*}$,
$B = B_1B_1^{*}$, where $*$ denotes adjoint.
Then we have $$C=A+B=\left( \begin{matrix}
A_1& B_1\\
\end{matrix} \right) \left( \begin{array}{c}
{A_1}^*\\
{B_1}^*\\
\end{array} \right) .$$
Reversing the order of the factors in the last product, and using the fact
that this reversal leaves unchanged the nonzero eigenvalues, we get that
$$\left( \begin{array}{c}
{A_1}^*\\
{B_1}^*\\
\end{array} \right) \left( \begin{matrix}
A_1& B_1\\
\end{matrix} \right) =\left( \begin{matrix}
{A_1}^*A_1& {A_1}^*B_1\\
{B_1}^*A_1& {B_1}^*B_1\\
\end{matrix} \right) $$
has $\gamma _1,\cdots ,\gamma _n,0,\cdots ,0$ as its eigenvalues.
Making a block diagonal unitary similarity to diagonalize the blocks $A_1^{*}A_1$ and $B_1^{*}B_1$, we can get that $\alpha _1,\cdots ,\alpha _n,\beta _1,\cdots ,\beta _n$ are diagonal elements of a Hermitian matrix having $\gamma _1,\cdots ,\gamma _n,0,\cdots ,0$ as its eigenvalues.
Then the author writes $$\left( \begin{array}{c}
\alpha _1\\
\vdots\\
\alpha _n\\
\beta _1\\
\vdots\\
\beta _n\\
\end{array} \right) =S\left( \begin{array}{c}
\gamma _1\\
\vdots\\
\gamma _n\\
0\\
\vdots\\
0\\
\end{array} \right) $$
where $S$ is a $2n$ -square doubly stochastic matrix.
My question is how to determine the existence of $S$.
I think there should be some relationship between the diagonal elements of the Hermitian matrix and its eigenvalues.
Maybe it is a theorem or I am not thinking enough.
I would like to get some references or suggestions.
Thanks!
Best Answer
The relevant theorem is the "$\implies$" direction of the Schur-Horn theorem. As the proof in the Wikipedia page demonstrates, a suitable $S$ can be constructed directly using the spectral decomposition of your size-$2n$ Hermitian matrix.
Notably, the majorization inequalities and equality referred to in the statement of the Schur Horn theorem are equivalent to the existence of such a doubly stochastic matrix $S$.