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Questions about the properties of vector spaces and linear transformations, including linear systems in general.
6,016 questions
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Spectral Clustering Algorithms Requiring Operator Diagonalizability
Let us focus on the category of directed graphs whose adjacency matrix (or random walk operator) is diagonalizable. Is there any spectral clustering algorithm that is specifically designed to operate ...
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2
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Matrices whose Lie commutator is the identity
Let $A,B$ two $n\times n$ matrices over a field $F$ and consider the usual Lie commutator $[A,B]=AB-BA$. As $[A,B]$ has trace zero, if the characteristic $p\geq 0$ of $F$ does not divide $n$ then $[A,...
- 720
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Characterization of positive definiteness of product of two positive definite matrices with special structures
Let $A$ be $n\times n$ matrix with special structure $A:=I_n+a\cdot \mathbf{1}\mathbf{1}^{\rm T}$, where $I_n$ is $n\times n$ identity matrix, $a>0$ is a scalar and $\mathbf{1}$ is an $n$-...
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Smallest eigenvalues comparison between two matrices: Seeking proof ideas
This problem ?stems from? a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the ?similarity transforming matrix from ${\bf J}$ to ${\bf ...
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113
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Calculations involving matrix exponential
I've come upon some calculations in a paper involving the exponential of a matrix that I'm confused by. I wonder if someone can help shed some light on this. As far as I can tell, for these ...
- 241
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Lower bounds on the eigenvalues of $X^T X$ when $X$ has bounded rows
Is there a way to find lower bounds on the eigenvalues of the matrix $X^T X$ that depend on the $\ell^1$ norm or the $\ell^2$ norm of $X$'s rows similar to this: Upper bounds on eigenvalues of PSD ...
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"Relative" Tits building for a subspace (for $\mathrm{GL}_n$)
I'm certain the following type of thing exists in the literature on buildings, since it seems not so hard to work out, but I'm not sure what the correct words are: the Tits building of $\mathrm{GL}_n(...
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On the Utility of Fractional Discrete Fourier Transform (FrDFT)-Invariant Signals
What are the theoretical and practical utilities of signals that are invariant under the Fractional Discrete Fourier Transform (FrDFT), specifically in real-world scenarios?
While the analogous ...
- 4,126
6
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1
answer
128
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Nice Riemannian metric with respect to mutually clean intersecting submanifolds
Suppose $M$ is a smooth manifold and $S_1, \ldots, S_k$ are smooth submanifolds which intersect mutually cleanly. This means for any subset $I \subset \{1, \ldots, k\}$, $$S_I:= \bigcap_{i\in I} S_i$$
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Why does the spectral gap tame a $K$-Lipschitz non-linearity on graphs?
I'm working on a problem involving states on a graph, where node states are transformed by a non-linear function. I need to verify a specific inequality related to the graph's spectral properties.
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1
answer
73
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Submatrix of a blowup of a Cayley graph adjacency matrix resembling an abelian Cayley graph?
Let $G = \operatorname{Cay}(\Gamma, S)$ be a Cayley graph over an arbitrary finite group $\Gamma$, and let $A \in \{0,1\}^{n \times n}$ be its adjacency matrix. For any positive integer $m$, define $...
- 431
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60
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submatrix of tensor product of matrices
Let $M$ be a zero-one $m$-by-$n$ matrix. Let $G$ be a graph whose vertices are pairs of row and column indices $(i,j)$ such that $M_{i,j}=1$, and $(i,j)$ and $(i',j')$ are adjacent if and only if $M_{...
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Integral Bruhat factorisations
Let $R$ be a principal integral domain (or even $R=\mathbb{Z}$ if it helps).
Let $M \in GL_n(R)$.
Say that $M$ has an integral Bruhat factorisation if $M=U_1 P U_2$ with $U_i$ upper triangular ...
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80
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Adjacency matrix of a Cayley graph on abelian group
Let $A$ be a symmetric $n \times n$ $\{0,1\}$-matrix with zero diagonal. I would like to understand when $A$ is the adjacency matrix of a Cayley graph over a finite abelian group.
That is, when does ...
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Sum of ranks of blocks
Let $M = \pmatrix{A & B\\ C& D}$, where $A$ is an all-one matrix. From Section 3 of Nisan & Wigderson$\color{magenta}{^\star}$,
$$\operatorname{rk} (B) + \operatorname{rk} (C) \le \...
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