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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higher Casimirs for $\mathfrak{sl}$

The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
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Which inner products preserve positive correlation?

Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
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Block matrices and their determinants

For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows: (a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...
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An inequality concerning the solution of a Lyapunov equation

Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
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Is $-\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2}$ always a square for each prime $p\equiv 3\pmod 4$?

Let $p$ be an odd prime and let $S_p$ denote the determinant $$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$ with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my ...
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Name of a matrix with one column and row removed [closed]

I am looking for the exact name of a matrix where the i-th column and rows have been removed. I cannot remember how it is called in linear algebra, does anyone got an idea? Thanks!
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An eigenvalue of certain family of matrices

Consider the matrices $$M_n=\left[\binom{i}j+\binom{2n+1-i}{j-i}+\binom{2n+1-i}j\right]_{i,j=0}^n.$$ I am convinced and hence would like to ask: Question: Is $0$ an eigenvalue of $M_n$?
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Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
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Bounds of Procrustes problem

We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following: \begin{align} f(A,B)...
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Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues

Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every ...
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Matrices with half columns nonnegative and half columns nonpositive real numbers

Suppose I want to study square even dimensional matrices that have the following property: Half of the columns have nonnegative entries and the other half have nonpositive entries. Now given an ...
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Walks of odd Lengths in a Matrix

Consider the following matrix $$A=\left[ \begin {array}{cccc} 1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0 \end {array} \right].$$ Assume that $B=A^k$ ...
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How to find the analytical representation of eigenvalues of the matrix $G$?

I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
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Computing spectrum of convex combination of SPD matrices given individual spectral decompositions

Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
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On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $$\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}$$ is a group ...
Consider the function $f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$ In the following we want to study the ...