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        Questions tagged [linear-algebra]

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

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        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 $\textbf{R} := (\textbf{I} - \textbf{M})^{-2}$,...
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        22 views

        Linear Functions question [on hold]

        Linear Functions, Determining The Equation Determine the equation of the line passing through the each point below having the corresponding slope. ( ? 4 , ? 4 ) The slope does not exist Y=-4 ...
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        1answer
        47 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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        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 ...
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        46 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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        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 ...
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        257 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 ...
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        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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        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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        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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        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 ...
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        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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        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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        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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        73 views

        Generalized Eigen property of a matrix

        Given a $n \times n$ invertible matrix $A$, I am interested in the set $$ \mathcal{S} = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}. $$ Thus for all eigenvalues $\lambda_i$, we have $\...

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