Answer 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 Article子育てに関する情報交換会 2023年5月27日(土) 10:00〜11:30
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