I am trying to prove the matrix inequality which came from the Gelbrich distance. The inequality seems to be correct as I substituted some random values, but not 100% sure with that. The inequality is as follows:
For the symmetric positive definite matrix $X$,$Y$,$M$,$N$,$A$,$D$
$|| \sqrt{X} - \sqrt{Y} ||_2^2 \geq || \sqrt{M} - \sqrt{N} ||_2^2$
where $M = \begin{bmatrix} A & B \\\ B^{\top} & D + X \end{bmatrix}, N = \begin{bmatrix} A & B \\\ B^{\top} & D+Y \end{bmatrix}$
Matrix 2-norm is defined as follows : $||X||_2^2 = trace(X^TX)$. The B matrix doesn't have to be square, and the only difference between M and N is the X and Y at the (2,2) position. The key point is that the size of the matrix between LHS and RHS are different.
I have tried some matrix inequalities such as Powers–Størmer’s inequality https://www.sciencedirect.com/science/article/pii/S0024379512006167, or https://mathoverflow.net/questions/361559/a-square-root-inequality-for-symmetric-matrices. But failed to prove it. It would be really helpful if anyone of you can give me some advice.
Have a good day, Thanks!