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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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      7
      votes
      3answers
      173 views

      Number of matrices with bounded products of rows and columns

      Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not ...
      5
      votes
      1answer
      186 views

      Inequality involving tensor product of orthonormal unit vectors

      Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
      2
      votes
      0answers
      121 views

      Is there literature on the study of “eigenmatrices”?

      Starting with a disclaimer: I will not be able to describe this with the correct terminology because I am trying to find the literature which I am unsure of it's existence. I will try to explain what ...
      1
      vote
      1answer
      42 views

      Rank and edges in a combinatorial graph?

      Fix a $d\in\mathbb N$ and consider the matrix $M\in\{0,1\}^{2^d\times d}$ of all $0/1$ vectors of length $d$. Consider the matrix $G\in\{0,1\}^{n\times n}$ whose $ij$ the entry is $0$ if inner product ...
      3
      votes
      0answers
      54 views

      Approximate inverse of large sparse matrix

      Given a large sparse matrix $M$, how to determine the existence of a good preconditioner? In other words, does there exist a sparse matrix $X$ such that $X M$ is close to the identity with respect to ...
      0
      votes
      0answers
      31 views

      Submatrix of uniform distribution on Stiefel manifold

      Let $U\in O(n,r)$ be uniformly distributed on the Stiefel manifold. Let $$X=\begin{pmatrix} U_{11}^2 & \cdots & U_{1r}^2\\ \vdots & \ddots & \vdots\\ U_{r1}^2 & \cdots & U_{rr}...
      1
      vote
      2answers
      259 views

      How many numbers in the matrix?

      We consider a matrix $\begin{bmatrix}a_{i,j}\end{bmatrix}$ with $r$ rows and $c$ columns. We fill this matrix only with zeros and ones. How many ones (maximally!) we can write into the matrix $r\...
      1
      vote
      0answers
      252 views
      +50

      Is this function quasi-concave?

      Let $$ \mathbf{u} := \left( \mathbf{X}^H \mathbf{X} + \mathbf{I}_m + \mathbf{\lambda}\mathbf{D} \right)^{-1} \mathbf{X}^H \mathbf{y} $$ where $\mathbf{X}$ is $n \times m$ semi-orthogonal matrix ($\...
      1
      vote
      1answer
      96 views

      Probability that random Bernoulli matrix is full rank

      This is probably known already, but I could not find a quick argument. Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the ...
      3
      votes
      1answer
      224 views

      Differentiability of operator norm [on hold]

      Is there any known results about differentiability properties of the function $\mathbb f:\mathbb R \to\mathbb R,$ $f(t):=\|A+tB\|_{op}$ where $\|.\|_{op}$ denotes the usual operator norm of the ...
      4
      votes
      0answers
      80 views

      Row rank and column rank of matrix with entries in a commutative ring

      Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed in "Ranks of Modules" one can say that the row rank of $A$ is ...
      0
      votes
      0answers
      51 views

      How to derive formula (10) norm to obtain formula (11) in Uncorrelated Group LASSO?

      In Uncorrelated Group LASSO, Eq. (10) and Eq. (11) are as follows: $J_2(W)=f(W)+\alpha Tr(W^TFW)$. (10) $F_{ii}=\sum_{g}\frac{(I_{G_{g}})_i||W_{G_g}||_{2,1}}{||W_{G_g}^i||_2}$. (11) where $w_{...
      7
      votes
      2answers
      164 views

      $2$-norm distance between square roots of matrices

      Suppose two square real matrices $A$ and $B$ are close in the Schatten 1-norm, i.e. $\|A-B\|_1=\varepsilon$. Can this be used to put a bound on the Schatten 2-norm distance between their square roots. ...
      3
      votes
      0answers
      126 views

      A combinatorial / geometric interpretation of compositional inversion via matrix inversion

      There are several ways of finding the power or Taylor series for the compositional inverse of a function $f(x)$ with $f(0)=0\;$ given its series expansion, e.g., by using the classic Lagrange ...
      0
      votes
      1answer
      66 views

      Energy of a symmetric matrix with $0$, $1$ or $-1$ entries

      I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...

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