All Questions

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 ...

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 $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases. My intention is to: decompose $\Phi$ via SVD: $U \Lambda V^T = \Phi$ to create it's ...