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I am trying to solve the following optimization problem for the vector $ y $, where $ A_i $ are some given matrix (maybe low rank) and $ x_i $ are unconstrained $$ \min_{y, x_i} \sum_{i=1}^J || y - A_ …

Let $A, B \in \mathbb{R}^{n \times m}$ be positive matrices. Consider the problem, $ \underset{A, B}{\arg\min} || A + B - T ||_F $ subject to $A > 0, B > 0$, and $(A - B) = O$ is orthogonal and $A_{ …

Here are alternate encodings based on Sebastian's comments on my previous answer. Comment: If you want to constrain the solutions found to a particular sign pattern matrix then an additional $mn$ li …

$$X_1 = RR' - x_1x_1'$$ U is unitary therefore, $$X_1 = RUIU'R' - \frac{1}{r}RUhh'U'R'$$ Let $M = RU$ then $X = MM'$ So the question becomes whether $MM' - M\frac{hh'}{r}M'$ is psd or not ? so we go t …

First consider the simple case, if $Q_1 = I$, then you want to find $Q_2$ and $R_2$ such that $Q_2 R_2 = \begin{bmatrix} R_1 \\ I \end{bmatrix}$. Split the columns of $Q_2^T$ into left and right halve …

Assume that $N$ is a real valued matrix. Let $x$ be an eigenvector corresponding to $\lambda_s$, i.e. $N_sx = \lambda_sx$. Note that $N_ax$ is always orthogonal to $x$. Therefore $||Nx||^2 = {\lambda_ …

Lets say that we have n matrices of data $X_i : i \in [1, n]$. All $X_i$ have the same number of rows. Their associated projection matrices are $P_i = X_i(X_i^T X_i)^{-1}X_i^T$ Also say that we have …

This is more of a comment but are you sure that this is a well-motivated problem ? Also this question is probaly not a good fit for MO (maybe stats.stack or math.stack?). What I mean is that usually …

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following: For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw > 0$. …

Just wanted to note that if one knows that the rank two update is of the form $uv^t + vu^t$ and one knows $v$ and $u$ then it is easy to find $x,y$ such that $uv^t + vu^t = xx^t - yy^t$. $x,y = \sqrt …

Here's one approach to solve this problem. Since $U$ is a thin matrix with orthogonal columns each of which has unit norm it spans a $d$ dimensional subspace within the $n$ dimensional ambient subspa …

p.s. - This is really a comment but I don't have enough rep. It seems, you want to bound $$ X_n = A_n^{-1} B_n $$. Let's say we simplify, that $A_n$ and $B_n$ are independent, and then start by assu …

$\epsilon = 0 \implies \exists v: v $ is eigenvector of N and $v$ is orthogonal to $e_1 \implies Mv = Nv \implies v $ is an eigenvector of $M$ with same eigenvalue. So if $\epsilon = 0$ then $\delta$ …

Assume your matrix is real valued. If we do $QR$ factorization of a matrix $A$, then $Q$ doesn't tell you anything about the kernel. It tells you about the range space of $A$. In fact the right-most …

Your problem has been answered at https://scicomp.stackexchange.com/questions/14096/sparse-smallest-eigenvalue-problem-on-a-linear-subspace :) Or you can read Golub's original paper Some modified mat …