# Tagged Questions

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### Source for roots of matrix polynomials?

A matrix polynomial is a polynomial whose variables are square $n \times n$ matrices, let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$. I am seeking a source of results on ...
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### dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...
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### Stabilization of the pencil of skew symmetric matrices by the orthogonal group

Good morning everybody. During my researches I've come across the following question. Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the ...
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### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
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### Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this? ...
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### Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...
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The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset ... 1answer 109 views ### The spectral radius of a modified graph Let$H$be a graph and let$G=H \vee K_{1}$be obtained by creating a new vertex and joining it to every vertex in$H$. This situation has many different names:$G$is called the cone or the ... 1answer 196 views ### Can sparse matrices satisfy the Null Space Property? Definition A matrix$A \in \mathbb{C}^{m \times N}$with$m < N$satisfies the Null Space Property (NSP) of order$s$if $$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ... 1answer 147 views ### A family of skew-symmetric matrices corresponding to cycles in graphs When investigating loops in Markov chains I ran into the following observation. A cycle in a graph G with n vertices may be represented by a matrix \Gamma \in \mathbb R^{n \times n} having the ... 3answers 156 views ### Applications of rank factorization or full rank decomposition [closed] I am teaching a course on linear algebra and came to this theorem: every m \times n matrix A with rank r admits a factorization A = CR where C is an m \times r matrix and R is an r ... 1answer 506 views ### SL_2(\mathbf{Z},8\mathbf{Z}) differs from E_2(\mathbf{Z},8\mathbf{Z}). Has this result appeared in the literature? I know a proof that the congruence subgroup SL_2(\mathbf{Z},8\mathbf{Z}) differs from its subgroup E_2(\mathbf{Z},8\mathbf{Z}), but can't find this fact in the literature. Does anyone know a ... 1answer 136 views ### Growth of powers of non-negative integer matrices In what I am currently doing, there naturally appears the following question: let A be a square matrix with non-negative integer entries. Let a_n be the sum of all entries of A^n. Question: How ... 1answer 333 views ### Rank of a 0-1-matrix Suppose K is a field of characteristic 0. Let M \in K^{n \times m} be a matrix such that every entry of M is either 0 or 1. About this matrix, I know further that each sum over a column ... 1answer 154 views ### What is the name of this measure of matrix “degenerateness” Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called? If the vectors are columns in a matrix \Phi, then ... 2answers 258 views ### tracial triples Say that a triple of real numbers (a,b,c) is a realizable triple if there are matrices A,B\in SL_2(\mathbb{R}) such that tr (A)=a, tr (B)=b, and tr (AB)=c. Question: what is the shape of the ... 6answers 1k views ### how to find/define eigenvectors as a continuous function of matrix? I asked this (with background) here http://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision but did not really get any answers. ... 1answer 183 views ### Decomposition of Matrix to its sub-matrix with constant rank When we study the structure of simple graphs with a lot of 1 or -1 as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ... 1answer 95 views ### Infinite Real Symmetric Toeplitz Matrix Reference I am looking for a good starting point (book or articles) for studying Toeplitz matrices. Specifically as mentioned in the title, I am most interested in the case where they are of the form$$A = ... 0answers 95 views ### “Bell” or “Jabotinsky”-matrix - What's the canonical name (if any)? I'm just reading J. Cigler's script for his talks "Konkrete Analysis" where I find the term "Jabotinsky-matrix" for that matrix, which I've (informally) been taught to call "Bell-matrix" (see at least ... 1answer 196 views ### Generating of the matrix ring by two hermitian matices Let$p$be a prime and$q=p^n$. Let$\mathbb F_{q^2}$be a field with$q^2$elements and$\sigma$its authomorhism of order two. A$m$by$m$matrix$A$over$\mathbb F_{q^2}$is hermitian if ... 3answers 397 views ### On certain decomposition of unitary symmetric matrices This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here. It is well known that a symmetric matrix over ... 0answers 130 views ### Does the following characterization of subgroups of$GL_2(\mathbb{F}_p)$generalise? Let$p$be a prime number. By a Cartan subgroup of$GL_n(\mathbb{F}_p)$I mean an absolutely semisimple maximal abelian subgroup. When$n=2$, it is well-known* that, for$G \subset ...
I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic. In 1953, Samuelson asked the following: If the ...