Articles of matrix equations

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 & C_{22} \end{pmatrix}\begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}$, where $\xi_1$ and $\xi_2$ have $p_1$ and $p_2$ dimensions, respectively. This problem has $p_1+p_2$ eigenvalues: $\{\rho_1, -\rho_1, …, \rho_p, -\rho_p, 0,…., 0\}$, where $p=min\{p_1, p_2\}$. Would somebody […]

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It’s the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn’t posed correctly and ended up generating responses that may be helpful for some people who stumble upon the question, but don’t address […]

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the spectral radius of $T$, $\rho(T)=\max\{|\lambda|: \exists v, Tv=\lambda v\}\leq 1$ and $|z|<1$ and $w$ is a primitive $n$th root of 1,i.e. $w =e^{i2\pi/n}$. I have tried using the identity $$[I-A]^{-1} = I+A+A^2+A^3+… $$ which holds when […]

Finding the basis of $\mathfrak{so}(2,2)$ (Lie-Algebra of $SO(2,2)$)

I want to calculate the Basis of the Lie-Algebra $\mathfrak{so}(2,2)$. My idea was, to use a similar Argument as in this Question. The $SO(2,2)$ is defined by: $$ SO(2,2) := \left\{ X \in Mat_4(\mathbb R): X^t\eta X = \eta,\; \det(X) = 1 \right\} $$ (With $\eta = diag(1,1,-1,-1)$) With the argument from the link, i […]

Solution of $A^\top M A=M$ for all $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can’t find out it there are other such matrices and if so, how to find them. Necessarily, $A$ is invertible since $\det(A)^2\det(M)=\det(M)\neq 0$ and more precisely, $\det A=\pm 1$. But […]

Proving that $p = \inf\{\|Ax-b\|: x\in\mathbb{R}^n\}$ is attained

I’m trying to solve the following problem in my textbook: Let $A$ be an $m \times n$ matrix of unspecified rank, $b\in\mathbb{R}^n$ and let $p =\inf\{\|Ax-b\|: x\in\mathbb{R}^n\}$ (the norm is abitrary on $\mathbb{R}^n$). Show that this infimum is attained (meaning, proving the existence of an $x$ for which $\|Ax-b\| = p$). I’m having a lot […]

Maximize trace of matrix equation given two constraints

Let $\mathbf{Q}$ be a rotation matrix and $\mathbf{A}$ and $\mathbf{B}$ be two real-valued matrices of the same size. I want to maximize the function $$ f(\mathbf{Q})=tr\;\mathbf{QA} \qquad \text{s.t.} \; \mathbf{Q}’\mathbf{Q} = \mathbf{I} \quad \text{and} \quad tr\; \mathbf{QB} \geq 0 $$ If only the orthogonality constraint is imposed, the solution is $$ \mathbf{Q} = \mathbf{VU}’ $$ […]

Clay Institute Navier Stokes

I’m trying to understand the Navier-Stokes Problem from the Clay Institute text here: . I’ve done an example problem to see what the problem is, but I don’t run into any. We are trying to satisfy: \begin{equation} \frac{\partial \textbf{u}}{\partial t} + (\textbf{u}\cdot\nabla)\textbf{u}=-\frac{\nabla P}{\rho} + \nu\nabla^{2}\textbf{u}+\textbf{f}, \end{equation} \begin{equation}\label{incompr} \nabla\cdot\textbf{u}=0. \end{equation} We get to pick a […]

An inequality on the root of matrix products (part 2 – the reverse case)

Suppose $A$ and $B$ are positive definite (symmetric) real matrices. In a previous post (An inequality on the root of matrix products) I asked whether $(AB)^{1/2}+(BA)^{1/2} \geq A^{1/2}SB^{1/2}+B^{1/2}S^TA^{1/2}$ ? where $S$ is a square contractive matrix (i.e. a square matrix that obeys $I-SS^T\geq 0$). This was shown by counterexample to be false in some cases […]

eigen values and eigen vectors in case of matrixes and differential equations

When I have an equation of the form $$a\frac{d^2x}{dt^2}+b\frac{dx}{dt}+c=0$$, then I can use Laplace transformation and transform it into $$as^2+bs+c=0$$ provided obviously$\quad x”(0)=0\quad \text{and}\quad x'(0)=0$. But the equation $as^2+bs+c=0$ is called the characteristic equation and the roots of the equation are called the eigen values $\lambda_1,\lambda_2$. Then we write the solution as $A\exp{\lambda_1t}+B\exp{\lambda_2t}$. I also […]