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I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references. The eigenvalue problem for a square matrix $M …

There is a lot of research about this. I can recommend the books Engl, Heinz Werner, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems. Vol. 375. Springer Science & Business Medi …

Sure, there is a lot about this, but maybe not under that name. You can find related ideas under the name of "best $k$-term approximation", for example in Ron DeVores Acta Numerica paper "Nonlinear Ap …

I don't know if this is what you want, but here are two differnet ways: In the case $m=n$: Generate a bunch of random permutation matrices $A_i$ and take a random linear combination $\sum_i \alpha_i …

Depending on the matrix free method: If it is iterative, you may just initialize it with a nonzero vector $x_0$. For example the Richardson iteration $x_{k+1} = x_k - A^T Ax_k$ does converge to the pr …

The SVD is used to analyze linear regularization methods for linear inverse problems. Here is very short introduction: A linear inverse problem is the challenge to find a good approximation of a linea …

$\newcommand{\RR}{\mathbb{R}}$Filtering an image $u\in\RR^{n\times m}$ by some filter $h\in \RR^{2r+1\times 2s+1}$ means computing $$ \sum_{k=-r}^r\sum_{l=-s}^s u_{i+k,j+l}h_{k,l} $$ at every pixel $( …

I only have an answer for the first question: A matrix $A$ with entries $A_{ij} = v_i+w_j$ is called on outer sum of $v$ and $w$. I would write this as $$A = v\oplus w := v\otimes 1 + 1\otimes w := v1 …

Phillip Lampe seems to be correct. Here are the eigenvalues and eigenvectors computed by hand: Let $k_1 = 2 + \tfrac12 + \cdots + \tfrac{1}{N-1}$, then: $\lambda_0 = 0$ with eigenvector all ones (by …

I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ an …

If you just ask about the relation between $A = USV^T$ and $\tilde A = U\tilde SV^T$ it is just that (for the spectral or the Frobenius norm) it holds that $$ \|A-\tilde A\| = \|USV^T - U\tilde SV^T\| …

The hairy ball theorem states that there is no nonvanishing continuous tangent vector field on even-dimensional spheres.

So you want to project $z$ onto the intersection of two convex sets $$C = \{x\mid \langle x, c\rangle \leq 1\}$$ and $$D = \{x\mid 0\leq x_i\leq 1\}.$$ The projection onto each of them is straightforw …

If you square your equations to get $|\langle x,a_i\rangle|^2 = b_i^2$ your problem is the so-called phase retrieval problem (which was motivated by the problem of recovering a function (up to global …

I do not know a definitive "weakest" condition and I doubt there is one. Many results in the realm of Hahn-Banach do the trick, i.e. there is the general result for $A,B$ convex and $A$ open (both ope …