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As @loupblanc states in his answer, there is ample literature on solving equations of the form $AXB+CXD=E$ in $O(p^3)$. Unfortunately, as far as I know, none of these methods extend to cases where the …

(EDIT: changing this to at least solve the stated problem...) We first look for the optimal vector for fixed $\|\lambda\|_1=M$. Hence the problem is: maximize $\|\lambda\|_2^2$ s.t. $\|\lambda\|_1=M$ …

This would probably get better answers on [scicomp.se]. Anyway: If you value accuracy over efficiency, using a QR factorization gives a backward stable algorithm, i.e., it guarantees that your soluti …

$ACB$ ranges over all the matrices with rank smaller or equal to the rank of $C$, so this is equivalent to a problem with $C=I$ (and possibly with a smaller $k$). That said, it is not clear to me how …

That problem does not have a maximum. Unless $A$ or $k$ are zero, you can take $\alpha = Me_j$, where $e_j$ is a vector of the canonical basis, and then the objective function diverges when $M \to \in …

Up to an orthogonal change of basis you may assume $U = \begin{bmatrix}I_d\\0\end{bmatrix}$. In this basis, the problem becomes $$ \min_{X_{11} \succeq 0} \left\|\begin{bmatrix}X_{11} & X_{12}\\ X_{12 …

Yes, they are connected. The first problem is a special case of the second when $\lambda=0$, if the first problem has a unique solution (i.e., $\ker A = 0$), or if the second is also formulated as a …

[EDIT: as OP confirm that they need the Frobenius norm, this should be a complete solution for that case] Since that norm is orthogonally invariant, $$ \|A - A'\| = \|P^T(A-A')\| = \|P^TA - DP^TA\| = …

Since you ask about positive entries, the Perron-Frobenius theorem limits possibilities. A possible characterization is the following: $$ A_1 = \{DMD^{-1} \colon \text{$D$ diagonal with positive $D_{i …

Using an SVD $A=USV^T$, and setting $y=U^Tx$, $c=U^Tb$, you can reduce the problem to a diagonal one $$\min_{y\in\mathbb{R}^n} c^Ty + \frac12 y^T S^2 y = \min_{y\in\mathbb{R}^n} \sum_{i=1}^n c_i y_i + …

The comments to this question of mine show that, in general, the operator-norm and Frobenius-norm minima are distinct for this problem. Let me summarize the argument here. Let $M=diag(\alpha,\beta,\ga …

If I am not missing something, this seems a direct application of Titu's lemma $$ \sum_{k=1}^K \frac{x_k^2}{y_k} \geq \frac{\left(\sum_{k=1}^K x_k \right)^2}{\sum_{k=1}^K y_k}, \quad x_k \geq 0, y_k > …

Answering to the edit: I still have no idea how $S$ is defined in terms of $G$. But that's fine, it can just be a $D^2\times D^2$ matrix for now. Not all matrices can be written as $S = U \otimes V$. …

A Kronecker product is essentially a rank-1 matrix, once you change indices: if $A_{ijkl} = U_{ik}V_{jl}$ (where all indices vary from $0$ to $n-1$, to keep the notation simpler), then the matrix defi …