Answer 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)...
$$\mathrm y^\top \mbox{diag}(\mathrm A^\top \mathrm B \,\mathrm A) \,\mathrm y = \sum_{k=1}^n \mathrm e_k^\top\mathrm A^\top \mathrm B \,\mathrm A \,\mathrm e_k \, y_k^2 = \sum_{k=1}^n \mathrm a_k^\top...
View ArticleAnswer by Rodrigo de Azevedo for Upper bound on the number of non-zero...
Rephrasing:Given matrices $\mathrm A_1 \in \mathbb R^{m \times k}$ and $\mathrm A_2 \in \mathbb R^{k \times n}$, is there an upper bound on the number of nonzero entries of the $m \times n$ matrix...
View ArticleAnswer by Rodrigo de Azevedo for Maximise singular value decay by sparse...
Maximizing the "decay" of the singular values could be thought of as minimizing the (numerical) rank. Hence, I believe that the original problem could be rephrased as follows:Given $\mathrm A \in...
View ArticleAnswer by Rodrigo de Azevedo for Positive definite matrices diagonalised by...
Let $n \times n$ matrix $\rm A$ be symmetric and positive definite. Since $\rm A$ is symmetric, it is diagonalizable. Hence, there exists a (non-singular) matrix $\rm P$ such that $\mathrm A = \mathrm...
View ArticleAnswer by Rodrigo de Azevedo for Calculate percentage of symmetry of a given...
Consider the following $2 \times 2$ matrix and its decomposition in a "natural" orthonormal basis.$$\begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 &...
View ArticleAnswer by Rodrigo de Azevedo for Solving system of bilinear equations
We have a system of $m$ bilinear equations in $\mathrm x, \mathrm y \in \mathbb R^n$$$\begin{aligned} \mathrm x^\top \mathrm A_1 \,\mathrm y &= b_1\\ \mathrm x^\top \mathrm A_2 \,\mathrm y &=...
View ArticleAnswer by Rodrigo de Azevedo for Determining if some permutation of a vector...
Let $\mathbb P_n$ be the set of $n \times n$ permutation matrices. Given matrix $\mathrm A \in \mathbb R^{m \times n}$ and vector $\mathrm v \in \mathbb R^n$, we would like to find a permutation matrix...
View ArticleAnswer by Rodrigo de Azevedo for Solving diagonal simultaneous quadratic...
We have the following system of quadratic equations in $\mathrm x \in \mathbb R^n$$$\mathrm A (\mathrm x \circ \mathrm x) + \mathrm B \mathrm x + \mathrm c = 0_m$$where $\mathrm A \in \mathbb R^{m...
View ArticleAnswer by Rodrigo de Azevedo for Minimization problem involving the inverse...
Rephrasing slightly, given (symmetric) matrix $\mathrm A \succeq \mathrm O_n$, we have the following minimization problem in (symmetric) matrix $\mathrm X \succeq \mathrm O_n$$$\begin{array}{ll}...
View ArticleAnswer by Rodrigo de Azevedo for Symmetric linear least-squares solution
Complementing Denis Serre's answer and rephrasing the original problem slightly, given tall matrices $\rm A$ and $\rm B$, we have the following quadratic program in square matrix $\rm...
View ArticleAnswer by Rodrigo de Azevedo for Applications of mathematics in clinical setting
From the Automatic Control Laboratory at ETH Zürich, a project on automating anaesthesia:The first steps to introduce feedback control in anesthesia were undertaken more than ten years ago. The project...
View ArticleAnswer by Rodrigo de Azevedo for Roots of determinant of matrix with...
Let $r_i := p_i - q_i$.$${\bf A} (x) := \begin{bmatrix} p_1 (x) & q_1 (x) & \ldots & q_1 (x)\\ q_2 (x) & p_2 (x) & \ldots & q_2 (x)\\ \vdots & \vdots & \ddots &...
View ArticleAnswer by Rodrigo de Azevedo for Applications of knot theory to...
Some papers on knots and proteins:William R. Taylor, A deeply knotted protein structure and how it might fold, Nature, Volume 406, pages 916–919, August 2000.Michael A. Erdmann, Protein similarity from...
View ArticleAnswer by Rodrigo de Azevedo for Prominent non-mathematical work of...
Doctor Ahmed Chalabi earned a PhD from the University of Chicago in 1969, founded Petra Bank in 1977, was sentenced in absentia in Jordan for bank fraud in 1992, and (allegedly) very successfully...
View ArticleAnswer by Rodrigo de Azevedo for Complexity of convex quadratically...
Convex quadratically constrained quadratic programming (QCQP) can be reduced to semidefinite programming (SDP). Suppose that we are given the following convex QCQP in $\mathrm x \in \mathbb...
View ArticleAnswer by Rodrigo de Azevedo for Mindset to understand category theory
You may want to take a look at Applied Compositional Thinking for Engineers (ACT4E), an ongoing graduate course at ETH Zürich:In many domains of engineering it would be beneficial to think explicitly...
View ArticleAnswer by Rodrigo de Azevedo for Spectral radius of a rank-1 perturbation
Arguably, the simplest case is where the matrix $\bf A$ is symmetric and positive semidefinite (PSD) and ${\bf u} = {\bf v}$, which ensures that the eigenvalues of the rank-$1$ update are in $\Bbb...
View ArticleAnswer by Rodrigo de Azevedo for Spectral radius of a rank-1 perturbation
To complement Christian's comment, since the spectral radius is upper-bounded by the spectral norm,$$\begin{aligned} \rho \left( {\bf A} + {\bf u} {\bf v}^\top \right) &\leq \left\| {\bf A} + {\bf...
View ArticleAnswer by Rodrigo de Azevedo for Eigenvalues of $\operatorname{diag}({\bf v})...
The smallest eigenvalue can be found (approximately) via the following semidefinite program (SDP).$$ \begin{array}{ll} \underset {t} {\text{maximize}} & t \\ \text{subject to} &...
View ArticleAnswer by Rodrigo de Azevedo for What phenomena are better modelled by SDE...
Since the Earth is not at zero degrees Kelvin, thermal noise is a reality that cannot be ignored. Much of electrical engineering is designing analog filters to remove as much noise as possible from...
View ArticleAnswer by Rodrigo de Azevedo for Is there a name for matrices of the form...
Some call them currency exchange matrices. From Boyd & Vandenberghe's Introduction to Applied Linear Algebra:6.7Currency exchange matrix. We consider a set of $n$ currencies, labeled $1,\dots,n$....
View ArticleComment by Rodrigo de Azevedo on Condition number can be arbitrarily worse...
Goes to infinity as what goes to infinity? Where is the convex optimization?
View ArticleAnswer by Rodrigo de Azevedo for Least-norm solution of the matrix equation...
Given symmetric matrix $\bf A$ and invertible square matrix $\bf B$, we have the following least-norm problem in square matrix $\bf X$$$\begin{array}{ll} \underset{{\bf X}}{\text{minimize}} & \|...
View ArticleOn computing the condition number of SPD matrices via convex optimization
Suppose that we have an $n \times n$ symmetric positive definite (SPD) matrix $\bf Q$ and that we would like to compute its condition number via convex optimization. In section 3.2 of Boyd et...
View ArticleAnswer by Rodrigo de Azevedo for An inequality for certain positive-definite...
$$ \tilde{\bf G} := \begin{bmatrix} 1 & \,\,\, {\bf a}^\top \\ {\bf a} & {\bf G} \end{bmatrix} $$where ${\bf G} \succ {\bf O}$ is a correlation matrix and ${\bf a} \geq {\bf 0}$. Since ${\bf G}...
View ArticleAnswer by Rodrigo de Azevedo for Bounding the norm of the inverse of a...
Given the invertible matrix ${\bf A} \in {\Bbb R}^{n \times n}$, we have the linear system ${\bf A} {\bf x} = {\bf 1}_n$, whose (unique) solution is denoted by ${\bf x}_0 := {\bf A}^{-1} {\bf 1}_n$. We...
View ArticleComment by Rodrigo de Azevedo on Unbiased estimator of the singular values of...
You might want to take a look at Alex Gittens's Random methods for Linear Algebra (2009) [PDF]
View ArticleComment by Rodrigo de Azevedo on Slight skew in the distribution of...
Normally distributed means $\sim \mathcal N (0,1)$?
View ArticleComment by Rodrigo de Azevedo on How do the singular values of a Hankel...
Where does this problem come from?
View ArticleAnswer by Rodrigo de Azevedo for INVERT transform as determinants of certain...
Let the HessenbergToeplitz matrix-valued function ${\bf M}_n : {\Bbb R}^n \to {\Bbb R}^{n \times n}$ be defined by$$ {\bf M}_n ({\bf a}) := \begin{bmatrix} a_1 & a_2 & a_3 & a_4 & a_5...
View ArticleComment by Rodrigo de Azevedo on Matching matrix columns under scaling,...
Do you know how to put the centroid of the columns of, say, $\bf Q$ at the origin? Right-multiply it by the rank-$(n-1)$ projection matrix ${\bf I}_n - \frac1n {\bf 1}_n {\bf 1}_n^\top$
View ArticleComment by Rodrigo de Azevedo on Designing a loss function in the form of a...
Do you agree with my edits?
View ArticleComment by Rodrigo de Azevedo on On the expected quality of a rank-$1$...
Instead of worrying about the phase, how about optimizing over $x_i \in {\Bbb C}$ subject to $x_i^* x_i = 1$ or $x_i x_i^* = 1$?
View ArticleComment by Rodrigo de Azevedo on Proving $\rho_{A}> \rho_{B}$ analytically
The largest real eigenvalue is not always the spectral radius. Should be the largest real eigenvalue in absolute value
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