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I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that: $$AX = B$$ Given that $X$ is ...

I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have $$\frac{\partial}{\partial A_{ij}} a^T(A^TA)^{-1/2}...

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently? $$(D + ADA^T)^{-1}$$ Hints or references are ...