Comment by Rodrigo de Azevedo on Symmetric linear least-squares solution...
In general, ${\bf X}^+ {\bf X} \neq {\bf I}$. Perform some experiments with MATLAB using pinv
View ArticleComment by Rodrigo de Azevedo on Symmetric indefinite matrix of fixed rank —...
You might want to take a look at slides 13-19 of Parrilo's The convex algebraic geometry of rank minimization (2009) [PDF]
View ArticleComment by Rodrigo de Azevedo on On a matrix equation with Kronecker product
If $A$ is not the unknown, why vectorise, then? Why not write in matrix form without $\otimes$?
View ArticleComment by Rodrigo de Azevedo on Solve NP-hard type problems with linear...
If you have a convex polytope whose vertices are integral, then you can solve integer programs via linear programming. But these are extremely special instances
View ArticleComment by Rodrigo de Azevedo on Fundamental regions in convex programming
Would "feasible" be better than "fundamental"?
View ArticleComment by Rodrigo de Azevedo on Norm bound in simultaneous stability to...
Is the operator norm the spectral norm?
View ArticleComment by Rodrigo de Azevedo on Two questions about three circulant matrices
Have you tried writing the $\binom{n+1}{2}$ equations in $3n$ binary unknowns for, say, $n=3$?
View ArticleAnswer by Rodrigo de Azevedo for Maximize the Euclidean norm of a matrix...
$$\mathrm A \mathrm x = \begin{bmatrix} \mathrm A_1 & \mathrm A_2 & \cdots & \mathrm A_n\end{bmatrix} \begin{bmatrix} \mathrm x_1\\ \mathrm x_2\\ \vdots \\ \mathrm x_n\end{bmatrix}$$where...
View ArticleAnswer by Rodrigo de Azevedo for When is the following block matrix invertible?
The only easy case I can think of is where $\mathrm A_{ij} = \mathrm A_0$, where $\mathrm A_0$ is an invertible $n \times n$ matrix, for all $i,j \in [d]$. In this special case,$$\mathrm A =...
View ArticleAnswer by Rodrigo de Azevedo for Nontrivial lower bound on the sum of matrix...
Let function $f : \mathbb R^{m \times n} \to \mathbb R_0^+$ be defined as follows$$f (\mathrm X) := \| \,\mathrm X \mathrm A^\top \|_\text{F}^2 + \| \,\mathrm X^\top \mathrm A \,\|_\text{F}^2 + \left(...
View ArticleAnswer by Rodrigo de Azevedo for Is this inequality involving the Frobenius...
Given $\mathrm A \in \mathbb R^{m \times n}$ and $\mathrm B \in \mathbb R^{n \times p}$, let $\mathrm B \mathrm B^{\top} = \mathrm Q \Lambda \mathrm Q^{\top}$ be an eigendecomposition of $\mathrm B...
View ArticleAnswer by Rodrigo de Azevedo for Quadratically constrained linear program...
We have the following (non-convex) quadratically constrained linear program (QCLP) in $\mathrm x, \mathrm y \in \mathbb R^n$$$\begin{array}{ll} \text{minimize} & \mathrm w^\top \mathrm x\\...
View ArticleAnswer by Rodrigo de Azevedo for Nearest matrix orthogonally similar to a...
Using the spectral norm, we have the following optimization problem in matrix $\mathrm X \in \mathbb R^{n \times n}$$$\begin{array}{ll} \text{minimize} & \| \mathrm A \mathrm X - \mathrm X \mathrm...
View ArticleAnswer by Rodrigo de Azevedo for Coupled Sylvester equations
We have two coupled linear matrix equations in $\mathrm X_1, \mathrm X_2 \in \mathbb R^{n \times n}$$$\begin{array}{rl} \alpha \, \mathrm A \mathrm X_1 + \mathrm X_1 \mathrm B &= \gamma \, \mathrm...
View ArticleAnswer by Rodrigo de Azevedo for Finding Toeplitz matrix nearest to a given...
The set of $n \times n$ symmetric Toeplitz matrices is$$\left\{ x_1 \mathrm M_1 + x_2 \mathrm M_2 + \cdots + x_n \mathrm M_n \mid x_1, x_2, \dots, x_n \in \mathbb R \right\}$$where $\mathrm M_1,...
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Answer by Rodrigo de Azevedo for Large power of an adjacency matrix
Since the adjacency matrix $\mathrm A$ is not symmetric, we have a directed graph.Given a positive integer $k$, the $(i,j)$-th entry of $\mathrm A^k$ gives us the number of directed walks of length $k$...
View ArticleAnswer by Rodrigo de Azevedo for How to calculate $y^T \mbox{diag}(A^T B A)...
$$\mathrm y^\top \mbox{diag}(\mathrm A^\top \mathrm B \,\mathrm A) \,\mathrm y = \sum_{k=1}^n \mathrm e_k^\top\mathrm A^\top \mathrm B \,\mathrm A \,\mathrm e_k \, y_k^2 = \sum_{k=1}^n \mathrm a_k^\top...
View ArticleAnswer by Rodrigo de Azevedo for Upper bound on the number of non-zero...
Rephrasing:Given matrices $\mathrm A_1 \in \mathbb R^{m \times k}$ and $\mathrm A_2 \in \mathbb R^{k \times n}$, is there an upper bound on the number of nonzero entries of the $m \times n$ matrix...
View ArticleAnswer by Rodrigo de Azevedo for Maximise singular value decay by sparse...
Maximizing the "decay" of the singular values could be thought of as minimizing the (numerical) rank. Hence, I believe that the original problem could be rephrased as follows:Given $\mathrm A \in...
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