All Questions

Given a matrix $L\in \mathbb{R}^{3 \times 3}$, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix: $$ \min_{s\in\...

The following problem is motivated from Generalized Procrustes Analysis. I am wondering if it is possible to obtain a closed form minimizer (which may involve SVD or some other decomposition of a ...

Given a nonzero vector $v \in \mathbb{R}^n$, we all know that it's projection onto the unit $\ell_2$ ball is just $\frac{v}{\|v\|}$. Let $X$ be some nonzero $n \times n$ matrix. What is the projection ...

Suppose $C_1$ and $C_2$ are some fixed $n \times n$ matrices. Define the norm $\| M \| = \sum_{i = 1}^n \max_j |M_{ij}|$. What is $\min_U \|C_1 U C_2 \|$? Here $U$ ranges over the $n \times n$ unitary ...

Context: I am trying to show the convergence of an optimization method which includes orthogonalization in the update step. Problem: Let's say I have real matrices $A, B \in \mathcal{R}^{n xm}$. If ...

Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...