I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two […]

I’m trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to $Rv=\{rv|r\in R\}$ for all $v\in M$. I’m trying to find a $M_n(D)$ module isomorphism $f:M_n(D)v\rightarrow D^n$. I’ve […]

If I have a set of matrices, call this set U, how can I make this a UFD (unique factorization domain)? In other words, given any matrix $X \in U$, I would be able to factorize X as $X_1 X_2 … X_n$ where $X_i \in U$ and this factorization is unique? We may assume the […]

I’m reading about the AHP method (Analytic Hierarchy Process). On page 2 of this document, it says: Given the priorities of the alternatives and given the matrix of preferences for each alternative over every other alternative, what meaning do we attach to the vector obtained by weighting the preferences by the corresponding priorities of the […]

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices. The geodesic distance between $A,B\in\Bbb{S}_{++}^n$ is given by the following Riemannian metric $$ d(A,B):= \Bigg(\operatorname{tr}\bigg(\ln^2\big(\sqrt{A^{-1}}B\sqrt{A^{-1}}\big)\bigg)\Bigg)^{\frac{1}{2}}. $$ EDIT It could also be defined as follows $$ d(A,B):= \lVert\log(A^{-1}B)\rVert_{F} $$ EDIT II I could show the negative-definiteness of $d^2$ if I could write […]

I’m looking for a classic matrix algebra reference, either introductory or advanced. In fact, I’m looking for ways to factorize elements of a matrix, and its appropriate determinant implications. Your help is greatly appreciated.

Let $A$ be an $n\times n$ symmetric positive definite matrix and let $B$ be an $m\times n$ matrix with $\mathrm{rank}(B)= m$. Show that $BAB'$ is symmetric positive definite.

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a matrix $Z=[\varepsilon_1, \varepsilon_2, \varepsilon_3, … ,\varepsilon_N] \in R^{n\times N}$. Using the information $||\varepsilon_i(x)||_2\le\gamma_0$, can I find an upper bound for $||Z||_2$? If […]

This question already has an answer here: $I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate] 3 answers

Let $W_1$ be the subspace of $\mathcal{M}_{n \times n}$ that consists of all $n \times n$ skew-symmetric matrices with entries from $\mathbb{F}$, and let $W_2$ be the subspace of $\mathcal{M}_{n \times n}$ consisting of all symmetric $n \times n$ matrices. Prove that $\mathcal{M}_{n \times n}(\mathbb{F}) = W_1 \oplus W_2$. I couldn’t really figure out why […]

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