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      Questions tagged [linear-algebra]

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

      2
      votes
      2answers
      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 &...
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      votes
      0answers
      21 views

      Understanding Partial Derivatives of a Neural Network

      I have to compute the following double derivative: $$ \partial _{x_i} \nabla_W \sigma(f(W,x))$$ where $W = (W_1, W_2, \dots, W_L)$ is the set of weight matrices, $f(W,x)$ is a $linear$ neural ...
      0
      votes
      0answers
      14 views

      Solution to quadratically constrained QP as linear combination of eigenvectors

      Let $\textbf{M}$ be a symmetric $n\times n$ matrix with zero diagonal, the strong Frobenius property, and spectral radis $\rho(\textbf{M}) <1$. Define $\textbf{R} := (\textbf{I} - \textbf{M})^{-2}$,...
      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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      vote
      0answers
      45 views

      Infinitesimal matrix rotation towards orthogonality

      TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal. Setting In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...
      1
      vote
      0answers
      56 views

      Infinite products from the fake Laver tables-Now with no set theory

      We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have $2^{n}*_{n}x=x$, $x*_{n}1=x+1\mod 2^{n}$,...
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      votes
      0answers
      107 views

      How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

      Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
      6
      votes
      2answers
      332 views

      Constructivist defininition of linear subspaces of $\mathbb{Q}^n$?

      Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek). I (and ...
      0
      votes
      0answers
      44 views

      Tensor factorizations [on hold]

      In matrix level, if I have a parameter matrix $M$ with a shape of $d \times d$. In order to reduce parameter size, the matrix can be factorized into two matrix $A,B$ with shapes of $d \times k,k\times ...
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      votes
      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{...
      6
      votes
      0answers
      167 views
      +200

      On a problem for determinants associated to Cartan matrices of certain algebras

      This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
      9
      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\...
      3
      votes
      1answer
      114 views

      Determinant of an “almost cyclic” matrix

      Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let $\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...
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      votes
      0answers
      42 views

      How to infer information about order of matrix [on hold]

      A and B are matrixs of n order, we know A^2+B^2=AB, and AB-BA is invertible, please prove the number of order n is multiple of three. I know a matrix is uninvertible when its determinant is zero, but ...

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