Articles of matrices

Ratio of area formed by transformed and original sides of a parallelogram

I am interested in finding the ratio of area formed by transformed and original sides of a parallelogram, given by: $$\frac{\|Ma\times Mb\| }{\| a\times b \|}$$ $M$ is a $3 \times 3$ matrix and $ a, b$ are vectors with 3 components each ($a,b$ are sides of the original parallelogram and $Ma, Mb$ are sides […]

Non-linear matrix equation solvable with linear algebra?

Consider the matrix equation $${\bf X}^k\bf {A = B}$$ Which we want to solve for $\bf X$ We can put A and B in a “block-vector”: $v = [{\bf A}^T,{\bf 0},\cdots,{\bf 0},{\bf B}^T]^T$, assume there exists a matrix: $$M = \begin{bmatrix}\bf I&\bf 0&\bf 0&\cdots&\bf 0\\\bf X_1&\bf 0&\bf 0&\cdots&\bf 0\\\bf 0&\bf X_2&\bf 0&\cdots&\bf 0\\\vdots&\ddots&\ddots&\cdots&\bf 0\\\bf 0&\bf […]

power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with entries following $|X_{ij}|\le1$. If $z$ has a small modulus ($|z|\ll1$), am I allowed to expand in a power series the square root matrix expression […]

Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then we consider another map $g:M_n(M_n(R))\rightarrow M_{n^2}(R)$ sending, e.g. $$\begin{pmatrix} \begin{pmatrix}1&0\\0&1\end{pmatrix}&\begin{pmatrix}2&1\\3&0\end{pmatrix}\\ \begin{pmatrix} 0&0\\0&0 \end{pmatrix}&\begin{pmatrix} 2&3\\5&2\end{pmatrix} \end{pmatrix}$$ to $$\begin{pmatrix}1&0&2&1\\ 0&1&3&0\\ 0&0&2&3\\ 0&0&5&2\end{pmatrix}.$$ Is it true […]

Finding only first row in a matrix inverse

Let’s say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what’s the best algorithm to use to calculate just this row? My matrix $M$ has the following properties: All its entries describe probabilities, i.e. take on values […]

Jacobian of $A (A^\top X A)^{-1} A^\top$

Let $A\in\mathbb{R}^{n\times m}$, $n\geq m$, be a full column rank matrix, and consider the function \begin{align} f&\colon \mathbb{R}^{n\times n} \to \mathbb{R}^{n\times n}\\ & X\mapsto A (A^\top X A)^{-1} A^\top, \end{align} where $\bullet^\top$ denotes transposition. Assuming that $(A^\top X A)^{-1}$ exists, I’m interested in the computation of the Jacobian matrix of $f$, i.e. $$\tag{1}\label{a} \mathbf{J}[f] = […]

Non-zeroth power of a Jordan block for the eigenvalue $1$ is similar to itself

I’m trying to prove: If $J$ is a single Jordan block corresponding to an eigenvalue $\lambda = 1$, then $J^k$ is similar to $J$, where $k$ is a nonzero integer. Moreover, if $\lambda = 1$ is the only eigenvalue of a matrix $A$, then $A^k$ is similar to $A$. Thanks

Iteration matrix and convergence

Assuming $G_{JA}$ is an iteration matrix for Jacobi Algorithm: $G_{JA}(A) = I -D^{-1}A$ $D$ is the diagonal of $A$. The sufficient condition for convergence is spectral radius less than one($\rho(G) <1$). Now, what happen if the spectral radius is equal to 1? Is there a way to set the parameter(such as initial guess) that guarantee […]

(Still open) Convergence of a certain matrix product representing a class of piecewise linear dynamical systems.

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline b_i & B_i \end{array} \right ] \quad$ where $\quad \exists (\| \cdot \|_\triangle). \forall i. \| B_i \|_\triangle < 1$. Here, $\Omega: \mathbb{R}^n \rightarrow \{1,2,\dots,K\}$ has the […]

Finding the eigenvalues and a basis for the eigenspaces of a $3\times3$ matrix.

For the matrix $A \in M_{3\times3}(\mathbb{R})$ below, I need to find the eigenvalues and a basis for the corresponding eigenspaces: $$\begin{bmatrix}\ 1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4 \\ \end{bmatrix}$$ I have tried to use the formula $\det(I\lambda – A) = 0$ but I ended […]