# Tagged Questions

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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### A naive question about the double dual of a vector space

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor $$V\mapsto V^{**}/V$$ of the category of $K$-vector spaces? I asked a related question on Mathematics Stackexchange, but ...
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### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
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### Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by \begin{equation*} e_k(\vx) := \sum_{1 \...
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### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
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### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
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### Determinant inequality involving Hermitian, positive definite matrices

Question: Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question ...
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Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\... 0answers 551 views ### An elementary linear algebra problem Let K be a field, and let E be the algebra of n\times n matrices over K. Let V_0 and V_1 be the (left) E-modules of matrices of size n\times n_0 and n\times n_1. Let W \subseteq V_0... 0answers 651 views ### Path connected set of matrices? Consider the collection of n by n matrices$$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$where c_i\in \{0,1\}, P_i and Q_i are disjoint ... 0answers 231 views ### Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But.... Let M be an n\times n matrix over, oh, let's say an algebraically closed field for now. There have ... 0answers 455 views ### Maximal set on hypersphere that does not contain pairs of orthogonal vectors Let R be a region on a hypersphere. Each point A of the hypersphere is associated with a vector pointing to A and with origin at the centre of the hypersphere. So let me identify each point with a ... 0answers 416 views ### integral matrix of order p Hi everyone Let p be a prime number. I am interested to classify \{ A\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \} up to conjugacy. One reason to consider this problem is its relation to ... 0answers 698 views ### coordinate-free proof of transitivity of norms or traces Hello: Suppose A is a finite free B-algebra and B is a finite free C-algebra. Does anyone know a coordinate-free proof (i.e. without choosing bases) of the identity: N_{A/C} = N_{B/C}\circ ... 0answers 192 views ### Standard polynomials applied to matrices (bis) The standard polynomial in r non-commuting indeterminates x_1,\ldots,x_r is defined by$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\...
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This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
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### Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
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### Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
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### Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
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### Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
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### maximal subgroups of $GL_2(Z/p^kZ)$

Hello, is there any classification of proper maximal subroups of $GL_2(\mathbb{Z}/p^k\mathbb{Z})$ for $k>1$ (analogous to the one which exist for $GL_2(\mathbb{Z}/p\mathbb{Z})$)? Could you give ...
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### Inverse of a matrix with binomial coefficients

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...
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### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...
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Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that: $e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$ Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x ... 0answers 241 views ### A variant of an Eventown problem for modulo a prime number Consider the following problem, called the 'Eventown problem': In a town, residents can form different clubs. The town council establishes the following rules: 1) Every club must have an even ... 0answers 164 views ### Numerical linear algebra: how to compute$B^TC^{−1}B$efficiently Hi, my question is similar to this one. I have to compute$B^TC^{−1}B$, where$C$is a strictly positive definite$n\times n$matrix and$B$is$n\times m$. The matrix$C$is huge ($n$up to a ... 0answers 403 views ### Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits Let$Sym^2(V)$be the set of symmetric matrices of a real$n$-dimensional vector space$V$. Given an element$\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where$\lambda_1\leq\...
The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space. The space of Hermitian matrices forms a cone in this vector space $M_n$...
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry. Suppose we have a system of $k\leq n$ polynomials in \$\...