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        Questions tagged [matrices]

        Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

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        Convex hull of all rank-$1$ $\{-1, 1\}$-matrices?

        Consider the set $\mathbb{R}^{m \times n}$ of $m \times n$ matrices. I am particularly interested in properties of polytope $P$ defined as a convex hull of all $\{-1,1\}$ matrices of rank $1$, that is,...
        1
        vote
        1answer
        72 views

        Matrix of powers of pairwise differences

        Let $\underline{c}:=\left(c_1,\dots,c_n\right)$ be pairwise distinct complex numbers, and let $k$ be a non-negative integer. Define the matrix $A_{n,k}(\underline{c})$ to contain the $k$-th powers of ...
        4
        votes
        1answer
        132 views

        Inverse of a matrix and the inverse of its diagonals

        While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have $${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D}...
        1
        vote
        0answers
        119 views

        Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?

        Disclaimer: This might be an SE question, but I'm not quite sure... Thanks in advance! Setup So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
        2
        votes
        1answer
        77 views

        Discrete dynamical system and bound on norm

        Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following: Consider the dynamical system with $x_i \in \mathbb C^2:$ $$ x_{i} = \left(\begin{matrix} z &&...
        1
        vote
        1answer
        100 views

        Minimal value of matrix norm induced by a norm

        Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by $$ \| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}} $$ where the matrix $A$ is interpreted as an ...
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        16 views

        Is the Weak Popov Form of a matrix over a polynomial ring unique?

        The Weak Popov Form of a matrix is a type of normal form of the matrix. We can find some special properties by changing each matrix to this form by considering some elementary operations on the matrix....
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        28 views

        A recap of regularity of singular values as a function over M_n

        So the core of the question is the study of the function $$ s :M_n(\mathbb R) \mapsto M_n(\mathbb R)$$ $$A \rightarrow s_n(A) $$ where $s_n(A)$ is the greatest singular value of A. I know there has ...
        5
        votes
        1answer
        265 views

        Determining the primitive order of a binary matrix

        Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows $$ {\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&1&1\\ 0&0&\cdots&0&0&...
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        0answers
        22 views

        Determinant of the Sum of a Left Circulant and a nonsingular matrix over field of order 2

        For any matrix $A=(a_{ij})\in M_n(\mathbb{F}_2)$, denote $\bar{A}=(\bar{a_{ij}}),$ where $\bar{a}=0$ if $a=1$ and $\bar{a}=1$ if $a=0$. Suppose $n$ is a power of two. Let $A$ (left circulant) and $I'$ ...
        0
        votes
        1answer
        60 views

        Number of symmetric matrix with fixed margins

        I'm looking for the cardinality of the set of symmetric matrices with entries in $\mathbb{Z}^*$ (the nonnegative integers) and fixed margins $\mathbf{k}=(k_1,k_2,...,k_q)$. I've also seen them called ...
        3
        votes
        1answer
        55 views

        Dimension of fixed vectors of a semi-linear operator

        Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
        8
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        1answer
        232 views

        What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?

        Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
        5
        votes
        1answer
        161 views

        Perturbing a normal matrix

        Let $N$ be a normal matrix. Now I consider a perturbation of the matrix by another matrix $A.$ The perturbed matrix shall be called $M=N+A.$ Now assume there is a normalized vector $u$ such that $\...
        4
        votes
        2answers
        79 views

        Stable matrices and their spectra

        I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices. A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...

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