Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,681
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Necessary conditions for the existence of solution of Sylvester equation AX=XB
Let's consider square matrices $A_{n \times n}$, $B_{n \times n}$ and $X_{n \times n}$ with elements from $\mathbb{R}$. Could you tell me please, what would be the necessary conditions for the ...
- 37.3k
-1
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1
answer
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On the permanent dominance conjecture for symmetric group
The Lieb's permanent dominance conjecture states that the expression $$\frac{d_{\chi}^HA}{\chi(e)}\le per(A)$$ holds for all positive semidefinite matrices $A$, where $d_{\chi}^HA=\sum_\limits{\sigma\...
CommunityBot
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votes
2
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Dimension of commutative subalgebras of a central simple algebra
let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division $k$-...
CommunityBot
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Explicit formula for this distance between positive semi-definite matrices?
Let $A$ and $B$ in $\mathbb{R}^{d\times d}$ be positive semi-definite (psd) matrices and let $d\tau$ be the uniform probability distribution on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$. I ...
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Linear matrix recurrences with polynomial coefficients
I am interested in solving linear recurrences of the form
$$a_{n+1}=\sum_{i=0}^K n^i X_i + \sum_{i=0}^L n^i Y_i a_n \tag{1}$$
where the $Y_i$ are $N\times N$ matrices, and the $X_i$ and $a_n$ are $N\...
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votes
1
answer
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Infinite tensor products
Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
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Largest subset not spanning the span
Let $S=\{c_1,\dots,c_n\}$ be a set of vectors in $\mathbb{R}^M$. Is the below problem studied in literature?
$$\max\limits_{S'\subset S} \vert S' \vert $$
$$s.t. dim(span(S')) < dim(span(S))$$
...
2
votes
1
answer
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Upper-bounding $\dim \text{span}\{v_1,\dots,v_n\}$ in terms of $\dim \text{span}$ of subsets
I asked this question on Stack Exchange two weeks ago, and didn't get any answers, so I'm shamelessly reposting it here.
Let $S=\{v_1,\dots, v_n\} \subset V$ be a set of nonzero vectors in a vector ...
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0
answers
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Rank of Vandermonde matrices
Consider a Vandermonde matrix
$$V = \begin{bmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\
1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\
& & \vdots \\
1 & x_n &...
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0
answers
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Link between eigenvalues of a symmetric matrix and a functional space
Let $f_1,\dots,f_n \in L^2(\mathbb{R},\mathbb{R})$ be $n$ mutually orthogonal functions with $\int f^2_i =1$ such that $|\{x \in \mathbb{R} | f_i(x) = 0\}| = 0$ for any $i \in \{1, \dots,n\}$. Does ...
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Comultiplication of elements of partition of unity
Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra).
Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
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How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?
I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\...
CommunityBot
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$\epsilon$-net under Hausdorff distance
Consider linear subspaces of $\mathbb{R}^n$. For two subspaces $X$ and $Y$, we define their Hausdorff distance as
$$
{\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X, |x|_2=1}\inf ...
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Eigenvalues of tridiagonal symmetric matrix
Could you tell me please, are there any analytical methods how to find eigenvalues of matrix such this one?
$$
\begin{pmatrix}
a_1 & b_1 & 0 & 0 & 0 & \ldots & 0 \\
b_1 & ...
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5
votes
2
answers
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Lie's theorem in characteristic $p$
Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
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Precise probability that $m$ random vectors in $n$ dimensional space are nearly-orthogonal
Consider $m$ vectors $v_1,\dots,v_m$ in $\mathbb R^n$, drawn uniformly and independetly from unit sphere. It is pretty straightforward from Chebyshev inequality that
$$
\mathrm P (\forall i\ne j \ |...
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1
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Matrix smoothly parametrized by t has eigenvalues (0, $\lambda$), eigenvector $v$. Is $\lambda v$ smooth?
Let $C(t)$ be a symmetric, two-by-two real matrix whose entries are smooth functions of $t \in \mathbb{R}$. Suppose that $C(t)$ point-wise has eigenvalues $\lambda$ and $0$. Then $\lambda(t)$ is a ...
- 118k
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Minimax optimization of diagonal entries of function of matrix
Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...
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Vectors that satisfy $\sum_{i=1}^n Y_i X_i^\top = I$ and $\sum_{i=1}^n \frac{1}{p_i}Y_iY_i^\top = \Sigma(P)^{-1}$
Let $X_1,\dots,X_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell_2$ ball. Let $P$ be a vector in the probability simplex $\Delta_n$ with $P_i>0$ for all $i$. ...
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Is the Wasserstein kernel positive definite?
Define a point cloud $X=\{x_i\}_{1\leq i\leq n}$, for $x_i\in\mathbb R^d$. Define the Wasserstein kernel as $$W(X,Y)=\max_{T}\frac{1}{n}\sum_{kl}T_{kl}\langle x_k,y_l \rangle$$
where $T$ is any doubly ...
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1
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Homomorphism induced by the second exterior power of a linear map
Consider the map from $M(n, \mathbb Z) \rightarrow M(\binom{n}{2}, \mathbb Z)$ taking a matrix A to its second compound, i.e, $\bigwedge^2 A$.
Restricting this map to the invertible matrices we get a ...
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votes
3
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Representation theorem for matrices (reference request)
Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$...
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Eigenvalues of structured matrices
Let $A=(a_{i,j})$ be an $n\times n$ matrix with $a_{j,j+1}>0,\; 1\leq j\leq n-1,$ and $a_{j,j-2}>0,\; 3\leq j\leq n$, the rest of the entries are zeros.
Is the following fact known:
All ...
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3-uniform hypergraphs and their circuit space
So, I'll break this post into two questions. Both concern 3-uniform hypergraphs. A 3-uniform hypergraph $H=(V,E)$ consists of a set of vertices $V$ and a set of edges $E$, where each edge $e\in E$ is ...
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Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
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Non-existent matrices with "essential zeros"
Is there a non-constant continous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and matrices $A=\begin{pmatrix}
a_1 & 0\\
0 & a_2\\
\end{pmatrix}$ and
$B=\begin{pmatrix}
b_1 & 0\\
0 & ...
- 7,463
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votes
1
answer
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How flexible is the infinite-dimensional torus?
Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group.
Problem 1. Is it true that for ...
- 4,102
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0
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Analytic function whose derivatives and primitives are independent from a given set of countable cardinality
Let $L=(l_j)_{j\in\mathbb{N}}$ be a set of countably many independent real analytic functions on $[0,2\pi]$. Here and in the following, independent means that a function cannot be written as finite ...
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Smallest eigenvalue for large kernel matrix
I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
...
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Not unique eigenvalues in singular value decomposition
I have the following problem: I have a matrix $M\in \mathbb{R}^{3\times 3}$ and I consider two SVD's $U_1DV_1^T$ and $U_2DV_2^T$ of $M$ with $D = \mathrm{diag}(\lambda_1,\lambda_1,\lambda_2)$. ...
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1
answer
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Singular value decomposition of random rectangular matrices
Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).
What is the ...
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votes
1
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Do these surfaces intersect?
For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
- 59.8k
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votes
0
answers
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Neat expresion for an anti-symmetric matrix
Fix a column vector $\pmb{v}$ and consider the cross product $\pmb{v}^T\times\pmb{x}^T$ for any column vector $\pmb{x}\in\mathbb{R}^3$. One can write
$$\pmb{v}^T\times\pmb{x}^T=A(\pmb{v})\pmb{x}$$
for ...
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classify antiholomorphic involutions of projective space
On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [...
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Factorization of a real matrix into Hermitian x Hermitian. Is it stable ?
It is known (see Theorem 4.1.7 in R. Horn & C. Johnson) that every matrix $A\in M_n(\mathbb R)$ (real entries) can be written as the product $HK$ of two Hermitian matrices (complex entries). Of ...
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Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
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Are $n \times n$ special orthogonal matrices, all the entries of which have the same absolute value, possible for $n \neq 4$?
As I noted in my preceding question https://math.stackexchange.com/questions/3510189/give-a-general-class-to-which-a-specific-4-times-4-special-orthogonal-matrix
in equation (62) of their recent ...
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Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$
I would like to know what all the possible finite-dimensional representations of the following relations are.
$$A_0^3 = A_1^3 = A_2^3 = I \tag{1}$$
$$A_0 A_1^\dagger + A_1 A_2^\dagger + A_2 A_0^\...
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Is there a bound on the norm of the product of second moment matrix with random vector?
Let $X_1,\dots,X_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell_2$ ball, but outside the ball of radius $r$ for some $r \in (0,1)$, i.e. $r \leq \|X_i\| \leq 1$ ....
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Bar notation in Bourbaki’s *Lie groups*, Chap. IX
I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...
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Naive generalization of determinant from matrices to higher rank tensors
Recall that using the Levi-Cevita symbol the determinant can be written as
$$\operatorname{det} A=\frac{1}{n!}\epsilon_{i_1\dots i_n}\epsilon_{j_1\dots j_n}A_{i_1j_1}\dots A_{i_nj_n}$$
Some ...
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What is the average area of the shadow of a convex shape taken over all possible orientations?
If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape?
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votes
1
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Matrix positive semi-definite
We construct a non-random matrix using random variables as follows:
We fix the vector $v=(1,1).$
Let $X$ be a $\mathbb R^2$-valued random variable such that $X$ is distributed according to
$$d\mu(...
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Subspace proximity
Given two rank-$k$ projection matrices $\Pi_1, \Pi_2 \in \mathbb{R}^{n \times n}$ (i.e. $\Pi_1 = \Pi_1^2$ and $\Pi_2 = \Pi_2^2$), what is the best function $c_{n,r}$ such that $\|\Pi_1 - \Pi_2 \|^2_F \...
- 60.4k
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Partitioning the set of Pauli words into abelian pieces
Let $\sigma_x,\sigma_y,\sigma_z$ be the Pauli matrices. A Pauli word of length $n$ is defined as the tensor product $\otimes_{i=1}^n\sigma_i$ of operators $\sigma_1,\dots,\sigma_n\in\{\mathbf 1,\...
- 4,435
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Complexity of pseudo-inverse of random matrix
Assume that $\mathbf{A}_{M\times N}$ is a sparse complex matrix. Then, what is the complexity of computation of its pseudo inverse, i.e.,
$$\mathbf{A}^{\mathrm{H}}(\mathbf{A}\mathbf{A}^{\mathrm{H}})^{-...
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Applications of linear algebra in the design of aircraft [closed]
David Lay mentioned one application of linear algebra in the design of aircraft in the introductory part of chapter 2 of his book:
[...] A computer creates a model of the surface by first ...
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votes
2
answers
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Three half circles on the plane may not meet nicely
Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...
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1
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Orbits of tensor product $\operatorname{St}_2\otimes\operatorname{Sym}^2(\mathbb C ^3)$
Let $G_1=\operatorname{GL}_2(\mathbb C)$ act on $V_1=\mathbb C^2$ via the standard multiplication. Denote this representation by $\operatorname{St}_2$. Let $G_2=\operatorname{SL}_3(\mathbb C^3)$ act ...
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0
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Case of equality in entrywise spectral radius bound
Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that
$$\...