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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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      34 views

      Rate of convergence for eigendecomposition

      Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix $$\Delta_D := \left( \begin{matrix} 2 &...
      2
      votes
      1answer
      51 views

      Bounding entries of the inverse of a matrix with bounded entries

      Let $A$ be an $n$-by-$n$ matrix with integer entries whose absolute values are bounded by a constant $C$. It is well-known that the entries of the inverse $A^{-1}$ can grow exponentially on $n$. (See ...
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      votes
      0answers
      52 views

      Eigenvalues and eigenvectors of a very large sparse matrix

      I have three questions about calculating the eigenvalues and eigenvectors of a very large sparse matrix. First of all, I have some large sparse matrices, some of which have only two nonzero diagonals ...
      3
      votes
      1answer
      61 views

      Mapping inclusion theorem for the numerical range

      We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$. Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
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      0answers
      41 views

      The derivative in inverse matrix [on hold]

      I wonder how to calculate the following derivative w.r.t to matrix: $\frac{d(x^TW^{-T}W^{-1}x)}{dW}$, where $W$ is a $\mathbb{R}^{d\times d}$ matrix and $x$ is a $\mathbb{R}^d$ vector. The result ...
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      vote
      0answers
      31 views

      Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix

      Lets assume we have the following equation: $AU=\lambda U \Rightarrow\left[ \begin{array}{c|c|c} 0 &A_{12}&A_{13}\\ \hline A_{21}& 0& A_{23}\\ \hline A_{31}&A_{32}&0 \end{...
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      votes
      0answers
      148 views

      More mysterious properties of Gram matrix

      This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question. The following fact could be extracted from 0402087: For any $a_i\...
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      votes
      0answers
      148 views

      A curious $q$-identity

      Let $[x]_{q}=\frac{1-q^x}{1-q}$ and $\binom{x}{n}_{q}$ denote a $q$-binomial coefficient. Let $A_n(x,q)$ be the $n\times n $ matrix with entries $$q^\binom{i-j}{2}\binom{i+j+x}{i-j+1}_{q},$$ $0 \le i,...
      16
      votes
      1answer
      235 views

      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
      82 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 ...
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      votes
      1answer
      136 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}...
      2
      votes
      0answers
      126 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
      78 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
      103 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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      votes
      0answers
      19 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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