Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,873 questions
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Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix
Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
- 109
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143
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A Riemannian manifold with a non-degenerate metric and an inner product $u_{\beta} u^{\beta}=1$
The question is: given a Riemannian manifold with a non-degenerate metric g and an inner product $u_{\beta}u^{\beta}=1$, is $\nabla_{\mu} (u_{\alpha}u_{\beta})=0$ without demanding the trivial ...
- 9
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81
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Constructing set with maximal independent subset
What is the minimal $m$ such that there exists a set $A = \{a_1,...a_n\}$ of vectors : $a_i \in \{0,1\}^m$ ($n$ is given) such that every subset of vectors of size $k$ is independent, but only with ...
- 51
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59
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A system of inequalities involving a skew-symmetric integer matrix
Which skew-symmetric integer matrices $S$ satisfy the following inequalities
$SV_i \ne z_iE_i$ for all $i = 1,\cdots, n$
where
$V_i$ denotes the column with integer entries such that the $i$-th ...
- 356
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41
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Orthogonality condition of symmetric matrix pencil
Let $P(\lambda)=\lambda M−L\in \mathbb{R}^{n \times n}$ be a matrix pencil with symmetric nonsingular matrix $M$ and $L$ is a weighted Laplacian matrix of a connected graph. Clearly $(0,1_n)$ is an ...
- 21
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84
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A permutation statistic and determinantal identity
I'm trying to read this paper, Total positivity, Grassmannians and networks by Postnikov (https://arxiv.org/abs/math/0609764) and I'm stuck on Lemma 5.1, which is essentially an identity about maximal ...
- 1
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136
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Finding a specific solution to $X^T\Sigma X = D$
I'm looking to solve for a specific $X$ in the following equation:
$$X^T\Sigma X = D,$$
where $\Sigma \succ 0$, $D$ is a diagonal matrix with strictly positive entries, and all matrices are square. It ...
- 41
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99
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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 ...
- 31
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179
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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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94
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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 ...
- 43.2k
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287
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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.,
$$\mathbf{A}^{\mathrm{H}}(\mathbf{A}\mathbf{A}^{\mathrm{H}})^{-...
- 287
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84
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Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
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126
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Tauberian operators
Let $X$ be a Banach space non reflexive and $T$ from $l_2(X)$ to $l_2(X)$ a bounded operator defined by:
$$T(x_n )=\frac{x_n }{n}.$$
We know that :
$$T^{**-1}(l_2(X))=\{x_n^{**} \in l_2(X^{**}) : \...
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255
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Span of a nonlinear function
Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...
- 159
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352
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Spectral norm of difference of quadratic matrices restricted to a subspace
Say that we have two matrices $X$ and $Y$ of dimensions $(T \times N)$ with $N < T$ and $rank(X)=rank(Y)=N$. Furthermore, define a $(T \times k)$ dimensional matrix $D$ with $k<N$ and $rank(D)=k$...
- 41
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96
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Eigenvalues of the matrix obtained by letting some of the rows vanish, hoping for some inequality
Let $A$ be an $n \times n$ matrix. Let $A_k$ be the matrix obtained by keeping the first $k$ rows of $A$ fixed and substituting $0$ for the rows $k+1$ to $n$. To be precise, we write $A= [R_1...R_k, ...
- 1,467
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400
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Comparison of two similarity matrices
English is not my first language, so please excuse any mistakes.
I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item ...
- 1
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83
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Nullity of infinite matrix in row echelon form
For an $m \times n$ matrix $A$ in row echelon form, $\mathrm{nullity} (A)$ is equal to the number of columns that do not contain a pivot. Is this also true for an infinite matrix in row echelon form ...
- 283
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67
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Singular values and the chromatic number
What relation, if any, is there between the singular values of the adjacency matrix ( or possibly incidence matrix) of a simple graph and its chromatic number. Typically, do we have Hoffmann type, or ...
- 2,089
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195
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Paths in graphs as a vector space or matroid
If I have a simple graph $G$, and what to count the number of simple paths between two distinct vertices, can the paths be seen as independent sets of a vector space, or even somehow, a matroid? I ...
- 640
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121
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Closed form solution to an equation
Let $X \in \mathbb{R}^{n \times d}, w \in \mathbb{R}^d, y \in \{\pm 1
\}^{n}, \alpha \in (0, 1)$. Consider the equation
$$ X^{\top}(Xw-y)+\alpha \|w\|_{2}X^{\top}\operatorname{sign}(Xw-y)+\alpha\frac{...
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87
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Orthogonal functions and linear operators
Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions,
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y)
$$
where $\boldsymbol{\beta}...
- 69
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35
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Converting a vector in a cone statement to inequality constraints
I would like to convert the following condition for $x$
\begin{align}
x = N \lambda, \lambda \geq 0
\end{align}
to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$
\begin{...
- 1
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145
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A cyclic inequality for a real vector space
Consider a finite-dimensional vector space $V$ over $\mathbb{R}$. A set of $n$ points $(x_i,y_i)$ in $V \oplus V^*$ is called good if
$$
(x_1,y_1) +\dotsb+ (x_n,y_n) \geq (x_1,y_2) + (x_2,y_3) + \...
user143954
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129
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Hadamard $\ell_2$ sum of two symmetric positive semidefinite matrices
This is a follow-up question to this and this.
Let $A=(a_{ij})$ and $B=(b_{ij})$ be symmetric positive semidefinite $n\times n$ matrices such that all $a_{ij}\geq 0$, $b_{ij}\geq 0$ and $a_{ii}=b_{ii}...
- 175
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78
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How to solve a quadratic matrix equation?
\begin{equation}
\boldsymbol{\omega^H} \boldsymbol{G} \boldsymbol{\Theta^H} \boldsymbol{h_r} \boldsymbol{h_r^H} \boldsymbol{\Theta} \boldsymbol{G^H} \boldsymbol{\omega}=a\\
\boldsymbol{\omega^H} \...
- 171
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223
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Solving a nonlinear matrix equation
Consider the following nonlinear matrix equation:
$B=PX^{−1}AX$
where $B$ and $P$ are a $1\times n$ row vector and $A$ is a $n\times n$ matrix which are all strictly positive, and $X=diag(x_1,...,...
- 101
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How do I test two square matrices are transpose to each other if only the column vector summations are known?
Given two secret square matrices, say $\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\
{{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\...
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83
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Matrix decomposition in a specific form
Can we prove that for any real valued $d\times d$ matrix $A$, $A$ can be decomposed to finite product of such matrices
$$A=\prod_{i=1}^n (I+R_i)$$
where $I$ is the identity matrix and $\operatorname{...
- 1
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87
views
General term formula for sequences
Let $k_1,k_2,\cdots,k_n,\cdots$ be a sequence of known positive numbers. Define
$$
a_1:=k_1,\\
a_2:=C_2^2k_2+C_2^1k_1a_1,\\
a_3:=C_3^3k_3+C_3^2k_2a_1+C_3^1k_1a_2,\\
a_4:=C_4^4k_4+C_4^3k_3a_1+C_4^...
- 411
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141
views
Two commuting matrices over a commutative ring
I would like to know if there are results about the dimension of the Algebra generated by two commuting Matrices over a ring (as there are in the case of a Field).
The good news is that "my" ring is ...
- 337
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1
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262
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Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
user136156
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176
views
Smallest eigenvalues of block Kronecker product
Let $D \in \mathbb{R}^{n \times n}$ defined as
\begin{equation}
D := \begin{pmatrix}
1 & 0 & \cdots & \cdots & 0 \\
-1 & 1 & \ddots & \ddots & 0 \\
\vdots & \ddots &...
- 133
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231
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What matrix has only negative or zero real part for all the eigenvalues?
Say $X \in \mathbb{R}^{m\times m}$,
Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part?
What I conjecture
The following $X$ has only negative ...
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36
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What is the locus defined by those equations?
I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by
$\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$.
I know that if $\...
- 121
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58
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Quadrics over the univariate function field with discriminant of minimal degree
Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...
- 1,833
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50
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Examples of Binary Functions that Yield Regular Graphs with Invertible Adjacency Matrix
Question:
What are, provided their existence, examples of functions $f$ with the following properties:
\begin{align}f:& \ \mathbb{N}\times\mathbb{N}\ni(i,j)&\mapsto\ \quad\quad\...
- 13.2k
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643
views
A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074
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64
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Probability of collision of sums of vectors
Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function.
Does there exist a random matrix $R \...
- 321
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369
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Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
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224
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Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle
Consider the matrix
$$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$
where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
- 101
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181
views
Number of Symmetric matrices
Let $S_m(q)$ denote the space of all $m\times m$ symmetric matrices over the finite field $\mathbb{F}_q$ of size $q$. What is the number of matrices $A=(a_{ij})\in S_m(q)$ of rank at most $3$ and $a_{...
- 179
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184
views
Oja's rule gives unit eigenvectors
Does Oja's rule for normalized Hebbian learning always result in a unit eigenvector which corresponds to the largest eigenvalue? Or are there any specific conditions or assumptions under which this is ...
- 101
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0
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98
views
Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
- 10.4k
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282
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A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180
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0
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93
views
Changing Couplings of Discrete Random Variables
Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
- 329
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0
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100
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Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
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0
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290
views
Need any information about an affine lattice
Motivation - I was thinking about calculating the integrals from An interesting integral expression for $\pi^n$? using old plain Riemann sums. There, one needs integrating over that part of $[0,1]^n$ ...
- 18.4k
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0
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40
views
Question about a systematic row reduction algorithm for compressive sensing
Suppose a ''brute-force'' algorithm is designed to systematically select from the first $n$ columns of an $m \times (n+1),$ $m<n$ augmented matrix $G$ representing a consistent underdetermined ...
- 259
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89
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Show that a certain ratio of diagonal entries dominates a certain ratio of singular values
Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum ...
- 723