# Questions tagged [linear-algebra]

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

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### Positive and trace-preserving transformations with a common fixed point of full rank

The following problem which has been on my mind for a while now arises from the realm of quantum information involving quantum channels with a common fixed point of full rank, as well as majorization ...

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65 views

### Subspaces of real $n \times n$ matrices of dimension $O(n)$

The set of real $n \times n$ matrices forms a vector space over the reals. Given any set $S$ of $n \times n$ matrices, there is a basis $S' \subseteq S$ of size at most $n^2$ such that any $x \in S \...

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398 views

### Certain matrices of interesting determinant

Let $M_n$ be the $n\times n$ matrix with entries
$$\binom{i}{2j}+\binom{j}{2i}, \qquad \text{for $1\leq i,j\leq n$}.$$
QUESTION. Is this true? There is some evidence. The determinant $\det(M_{2n+1})...

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21 views

### Explanation for Dependency of Solvability of a System of Linear Equations on a Number Theoretic Property

The origin of this question is that I found a way to 'eliminate' vertex weights from weighted $K_n$ graphs, i.e. if one assumes that the weight $w_{ij}$ of edge $e_{ij}$ can be expressed as $\pi_i+\...

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438 views

### Is a matrix similar to its transpose over $\mathbb{Z}_p$?

Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...

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374 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 ...

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67 views

### A question related to (random) matrix factorization

Let $\mathbf{B}$ be an $m\times r$ binary matrix taking values in $\{0,1\}$, and assume that $m\geq r$. I am wondering what can be said about the recovery of $\mathbf{B}$ from $\mathbf{B}\mathbf{B}^T$....

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26 views

### A $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...

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285 views

### Conceptual explanation for curious linear-algebra fact in characteristic $2$

All matrices and vectors in this post have entries in the field $\mathbb{F}_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...

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112 views

### On $\det\big[x+\big(\frac{i^2-\frac{p-1}2!\,j}p\big)\big]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$

Motivated by Question 302323 and Question 317509, I have formulated the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$, there is a positive integer $...

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111 views

### On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$

I have made the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...

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162 views

### Determinant of “skew-symmetric” matrices

For $n\in\mathbb{N}$ and $m=\lfloor\frac{n}2\rfloor$, consider the $n\times n$ skew-symmetric matrix $A_n$ where each entry in the first $m$
sub-diagonals below the main diagonal is $1$ and each of ...

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33 views

### Spherical code for interesection of $k$-sparse vectors and unit sphere

Let us assume $X\in\mathbb{R}^{n\times d}, rank(X)=d$, integer $k\in\mathbb{N},k\ll d$, positive constant $0<\epsilon<1$, and $\mathcal{S}\subset \mathbb{R}^d$ denotes the unit sphere. We also ...

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46 views

### Show the spectral radius of a matrix is smaller than 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...

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38 views

### Linear transformation group of paraboloid in $\mathbb{R}^d.$

Suppose $P$ is a paraboloid in $\mathbb{R}^d.$ What is the group of all linear transformations $\mathcal{L}: \mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $\mathcal{L}(P)=P?$
Could someone please ...

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21 views

### Find the analytical form of the eigenvalue of a special sparse matrix

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...

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47 views

### When are “square spans” not transversal?

Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...

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28 views

### Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...

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66 views

### Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...

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155 views

### Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...

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69 views

### 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) = \|\...

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76 views

### A question on a special “metric”

Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...

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75 views

### Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...

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255 views

### Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums

The following question arose in some discussions recently as a misunderstanding of another problem.
Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...

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108 views

### Matrices with distinct columns

Let $M$ be an $n \times m$ matrix over $\mathbb{F}_2$ with no repeated columns, and suppose that $m \leq 2^{n-1}$: i.e., it is possible to have a matrix with fewer rows that still has $m$ unique ...

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204 views

### Lefschetz operator

Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2\mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...

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99 views

### Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic

This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller.
Let $\mathbb{N}$ denote the set of positive integers and for $n\in\...

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148 views

### $2$-adic valuation on $\mathbb Q (\sqrt{-15})$

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that
$-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ denotes ...

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189 views

### Covering the finite plane with lines

This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...

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222 views

### How to verify the characteristic polynomial? [closed]

I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?

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172 views

### Forcing scalar products to avoid prescribed values

Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition:
\begin{gather}
\text{For ...

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68 views

### Eigenvalues of non-negative block matrices

$B$ is a non-negative irreducible block matrix as follows:
$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...

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110 views

### Maximal number of $S_n$-conjugates living in a hyperplane

Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...

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76 views

### Generic deformation of matrix

Let $A(x)$ be a $m \times n$ matrix, whose entries are real polynomials $f_{i,j}:\mathbb{R}^S \to \mathbb{R}$. Denote the ith row by $f_i$ And let $rk:\mathbb{R}^S \to \mathbb{N}$ be the rank function ...

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142 views

### Linear subspaces in quadric hypersurfaces

Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.
Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,...

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70 views

### Separate the trivial partition by a linear hyperspace

Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that:
$\langle a,e\rangle=0$ and
for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\...

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44 views

### Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

This is cross post to the question at MSE.
Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also ...

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307 views

### Decide if a matrix is transposable

A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...

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26 views

### Approximate Simultaneous Diagonalization of Non-Hermitian Matrices

Let $A_1,A_2$ two $n\times n$ complex matrices. $A_1$ and $A_2$ are also non-normal, especially, non-hermitian and do not commute. I would like to find an invertible matrix $V$ such that
$$
\sum_{i=1,...

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197 views

### Determinant of a block matrix with many $-1$'s

For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...

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31 views

### Choice of residual function for least squares error minimization

Good morning,
I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data.
I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as
$K_{IC} = \sigma \sqrt{D} k_0(\...

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438 views

### $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...

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168 views

### The maximal size of intersection of two sets

Let $S=\{1,2,\cdots,2n\}$, and $S_i \subseteq S(i=1,2,\cdots,n+1)$ be $n+1$ subsets, each of which contains half of the $2n$ elements, namely $|S_i|=n$. Consider the following expression:
$$M=\max_{1\...

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194 views

### Matrix Inequality: Traces of $n$th powers

Let $A, B$ be matrices over $\mathbb{C}$ of the same dimensions (not necessarily square). With $'$ denoting conjugate-transpose, and tr the trace, show for $n\in\mathbb{N}$ that
$ 2\,\mathrm{Re}\, \...

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**1**answer

96 views

### About roots of the equation $det(A-\lambda B)=0$ [closed]

Let $A,B$ be square real symmetric matrics of the same degree $n \geq 3$ without common isotropics vectors (i.e. there is no a nonzero vector $x\in \mathbb R^n$ such that $x^TAx=x^TBx=0$).
Are roots ...

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71 views

### Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?

Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...

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139 views

### Bounding the eigenvalues of $B A B^T$ with the eigenvalues of $A$

Given a Hermitian positive semi-definite $n \times n$ matrix $A$ and a rectangular $m \times n$ matrix $B$, is there anything that can be said about the eigenvalues of the matrix $B A B^T$?
It seems ...

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65 views

### What is the most efficient path for a robot without turning radius?

I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...

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184 views

### Conjecture that relates matrix systems with some specific functions as solution sets

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...

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174 views

### Making binary matrix positive semidefinite by switching signs

Let $A \in \{\pm 1\}^{n \times n}$ be a symmetric matrix whose diagonal entries are $+1$. Let $f(A)$ be the smallest number of signs we need to change in $A$ so that it becomes positive semidefinite (...