# Questions tagged [linear-algebra]

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

3,659 questions
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### Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then $$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$ where we embed the ...
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### Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently? $$(D + ADA^T)^{-1}$$ Hints or references are ...
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### Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
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### Determining if some permutation of a vector satisfies a system of linear equations

Let $A$ be a matrix and $x$ a fixed vector. How can we determine whether or not there exists a permutation matrix $P$ such that $APx=0$? Does this problem reduce to anything well-understood?
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### invertible endomorphisms on a space of linear maps between finite-dimensional vector spaces

This is a linear algebra question that came up in my research, and I feel like there ought to be either a simple proof or a simple counterexample, but I have been unable to find either. Assume $V$ ...
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### Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D

I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation: $AXB + (AXB)^T + cX = D$ where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...
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### When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE. Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...
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Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means $${\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ... 0answers 77 views ### Decomposition of a Matrix by Sparse Matrices Let \mathbb{F} is a field. Consider an n \times n matrix \bf A over \mathbb{F}. \bf A is called sparse matrix over \mathbb{F} iff the number of non-zero entities of \bf A be at most ... 0answers 32 views ### find linear approximation of non-linear matrix transform [closed] I have a square matrix denoted as A and an element-wise square operator \sigma, such that \sigma(A)=a_{ij}^2,\forall i,j, a_{ij} is the ith row and jth column element of A. Is there exists ... 0answers 35 views ### Unstable convergence of a Poisson equation What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ... 1answer 150 views ### Eigenvalues of n matrices For i \in \{1,...,n\}, let A_i=[a^i_{jk}] be a symmetric matrix. For i \in \{1,...,n\}, we assume that a^i_{jj}=0 for all j and rank (A_i)=2. Then it has one positive and one negative ... 4answers 936 views ### If A,B are upper triangular matrices such that AX=XA\implies BX=XB for upper triangular X, is B a polynomial in A? A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here. ... 1answer 97 views ### Inertial decomposition of graphs The problem is this: given a graph G, to find a decomposition of G, i.e. a set F of vertex-disjoint proper subgraphs of G such that:$$\text{inertia}(G) = \sum_{H \in F} \operatorname{...
Let $g(x) = e^x + e^{-x}$. For $x_1 < x_2 < \dots < x_n$ and $b_1 < b_2 < \dots < b_n$, I'd like to show that the determinant of the following matrix is positive, regardless of $n$: ...