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Questions tagged [linear-algebra]

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

-3
votes
0answers
15 views

Unable to understand that why dot product of columns is not zero for vectors (2,-2,1) and (1,2,2,) [on hold]

I am taking a linear algebra course on Udemy and in topics of Orthogonal projection a topic of zero dot products comes up where instructor explains that dot products of columns (2,-2,1) and (1,2,2,) ...
0
votes
0answers
11 views

A right-inverse property of a nonlinear optimization problem

Disclaimer: This might be a silly question. However, after some days of thought, I could not find a clear/rigorous answer. So I decided to post it here. Let $Y\in\mathbb{R}^{n\times p}$ and $X\in\...
0
votes
0answers
16 views

An two-norm estimate for symmetric $k$-tensors

Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...
0
votes
0answers
17 views

Stability analysis with minimal spectral norm

Let $A \in \mathbb{R}^{n \times n}$ with $$ s(A) = \inf_{D \textrm{ is diagonal}} \| D^{-1} A D \|_2 > 1 $$ Does there exists $m \in \mathbb{N}^n$ and $z \in \mathbb{C}$ with $|z| > 1$ such ...
1
vote
0answers
16 views

How explicitly write a projective transformation between the conics over the univariate function field?

Consider the quadratic forms $$ Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2 $$ over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...
-1
votes
0answers
19 views

relevant probability for vector in multi-dimensional spaces [on hold]

is it possible to compare the probability that for example a 2D vector lies on top of an other 2D vector within the 2D space, with the event that a 3D vector lies inside a 2D plane of the same 3D ...
0
votes
1answer
35 views

Intersection between a line and a n dimensional parallelotope

Suppose that I have a line in an $n$ dimensional space described by $$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$ here $A$ is known and I want to find all the possible vectors $B$ ...
1
vote
1answer
63 views

Inequality for the operator norm of a product of matrices

I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that $$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
3
votes
0answers
35 views

Minimize Frobenius instead of Spectral norm via diagonal similarity

Given square matrix $A$. I am looking for a numerical solution for $$ s(A) = \inf_D \| D^{-1} A D\|_2, $$ where $D$ is a non-singular, diagonal real matrix. A numerical solution was here. However, ...
2
votes
1answer
100 views

counting invertible matrices [on hold]

Let $T$ be a subset of vector space $Z_2^n$ and $A$ be an element of $GL(2,n)$ means invertible matrices with entries $\{0,1\}.$ Let $T$ be invariant under A. It means for any $t \in T$, $tA \in T$. ...
5
votes
0answers
87 views

Matrix operations preserving the middle coefficients of characteristic polynomial

Let $n$ be a positive integer. For a ring $A$ and matrix $M \in \mathrm{Mat}_{n \times n}(A)$, let $\chi_{M}(t) = \det(M-t \operatorname{id}_{n}) = (-1)^{n}(t^{n} - \sigma_{M,1}t^{n-1} + \dotsb + (-1)^...
0
votes
0answers
50 views

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 ...
5
votes
1answer
288 views

A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$

On the basis of my computation, here I pose my following conjecture involving the cosine function. Conjecture. For any positive integer $n$, we have the identity $$\frac1{2n}\det\left[\cos\pi\frac{jk}...
0
votes
0answers
24 views

Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set \begin{equation} S_{\{1,2\}} := \{...
2
votes
1answer
79 views

volume of parallelotope in $L^2(\mathbb R).$ [closed]

Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product. Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g., $$\{ f(...
26
votes
1answer
505 views

Determinants of binary matrices

I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
0
votes
0answers
27 views

Invertible matrices T that commutate with the Jordan matrices A of size n [closed]

I'm supposed to find the set of invertible matrices T that commutate with the Jordan matrices A of size n. Through trial and error, I've found that tᵢ₊₁,ₖ = tᵢ,ₖ₋₁. I haven't found a way to prove ...
2
votes
1answer
172 views

Are Linear Maps resistant to Noise?

Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some ...
3
votes
0answers
124 views

Combinatorics question

Let $A = (a_{ij})_{1\le i,j\le h}$ be an $h$-by-$h$ non-degenerate upper triangular matrix with entry $a_{11} = 1$. Let $\Phi = \{\alpha_1,\alpha_2,\ldots,\alpha_d\}\subseteq \{1,2,\ldots,h\}=I$ be an ...
0
votes
1answer
57 views

Maximum of a sum of Gaussian functions

Consider the function which maps $\mathbb{R}^n$ to $\mathbb{R}$ \begin{align} f(x) = \sum_{i=1}^{n} b_i\phi_i(x) \end{align} where $\phi_i(x) = \exp(-\frac{||x-x_i||_2^2}{2})$ are Gaussian functions ...
3
votes
0answers
67 views

How to find the best similarity transformation between two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$? [duplicate]

Suppose I have two matrices $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times n}$. I want to know what's the best similarity transformation between these two matrices when we ...
3
votes
1answer
119 views

Endomorphisms of the p-adic group $(\mathbb Z_p,+)$

Does there exist an endomorphism of $(\mathbb Z_p,+)$ of finite order different of $x\mapsto\xi x$ where $\xi$ is a root of unity in $\mathbb Z_p$? Thanks in advance
7
votes
2answers
249 views

Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$

Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?
2
votes
2answers
51 views

Lower bound of positive entropies of automorphisms on tori

Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy $$ h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \} $$ where $\lambda$ goes through all eigenvalue of $A$ ...
3
votes
1answer
150 views

Volume of polyhedron

Given the following polyhedron: All the $n\times n$ matrices $\boldsymbol{X}$ with elements $x_{ij}\in(0,1)$ such that $$\boldsymbol{X}\cdot\boldsymbol{1}=\boldsymbol{r}, \boldsymbol{1}^T\boldsymbol{...
0
votes
0answers
62 views

Representation of symmetric group as Cremona transformations

Question from me and a colleague: Given a matrix \begin{equation} U = \begin{bmatrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix} \quad \text{with } U_{22} \neq 0, \end{equation} ...
0
votes
1answer
26 views

tensor stability of block-positive matrices

Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation) $\langle \psi |...
2
votes
1answer
47 views

Lie-algebra-like relation for totally symmetric 4-tensors

There are many totally symmetric real 4-tensors, $T_{ijkl}$, which satisfy the relation $$T_{ijmn}T_{mnkl} + T_{ikmn}T_{mnjl} + T_{ilmn}T_{mnjk} = c T_{ijkl}$$ with some constant $c$. By the way ...
4
votes
0answers
113 views

How to formulate supercommutativity in a characteristic free way?

I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...
6
votes
0answers
66 views

mean distance between subspaces

Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...
5
votes
1answer
278 views

$(AB)^+\approx B^+A^+$ for $B$ “fat” enough?

Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse. Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
-4
votes
0answers
145 views

On the determinant $\det[(i^2+dj^2)(\frac{i^2+dj^2}p)]_{1\le i,j\le(p-1)/2}$ with $p$ an odd prime

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. We define the determinant $D(d,p)$ by $$D(d,p):=\det\left[(i^2+dj^2)\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}...
0
votes
0answers
29 views

Eigenstructure and condition number of a block symmetric matrix

Consider a block, symmetric matrix $$ \begin{pmatrix} 0 & -A^T \\ A & C \end{pmatrix} $$ where $A$ and $C$ are two real positive definite matrices. What is the condition number of that ...
3
votes
0answers
76 views

Jacobian of the action of a matrix on a Grassmannian

I'm looking for a reference concerning a calculation found in Furstenberg's 1963 paper "Non-commuting random products". Lemma 8.8 of this paper states that if one takes a $d\times d$ invertible real ...
4
votes
3answers
254 views

Question about an inequality described by matrices

Let $A=(a_{ij})_{1 \le i, j \le n}$ be a matrix such that $\sum_\limits{i=1}^{n} a_{ij}=1$ for every $j$, and $\sum_\limits{j=1}^n a_{ij} = 1$ for every $i$, and $a_{ij} \ge 0$. Let $$\begin{equation} ...
7
votes
2answers
270 views

An upper estimate for $|\det(A+B)|$

If $A$ and $B$ are $n\times n$ matrices, then it easily follow from the definition of the determinant by sum over permutations, and from the Young inequality that $$ |\det (A+B)|\leq C(n)(\Vert A\Vert^...
0
votes
0answers
20 views

Fairly allocating heterogenous items

I'm trying to find literature on what I'm sure is a well-understood mathematical problem, but am struggling for terminology. Let's say I have a number of items each of which is either shiny or matte, ...
0
votes
0answers
40 views

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\...
2
votes
0answers
51 views

Orthogonal Matrices and Cosets (translates) of Linear Subspaces

Let $M_n(F_2)$ be the vector space of all $n\times n$ matrices over the finite field $F_2$. Let $O(n)\subset M_n(F_2)$ be the set of all orthogonal matrices and $W\subseteq O(n)$ be an affine subspace ...
8
votes
0answers
363 views

Prove the optimality of the following constant

Let $E$ be a complex Hilbert space. In (arXiv) it was shown that for $A=(A_1,...,A_n) \in \mathcal{B}(E)^n$ we have, $$\displaystyle\frac{1}{2\sqrt{n}}\|A\|\leq \omega(A) \leq \|A\|,$$ where $$ \...
4
votes
1answer
124 views

The minimum rank of a matrix with a given pattern of zeros

For real matrices $A=(a_{ij})$ and $B=(b_{ij})$ of the same size, I write $A\prec B$ if $a_{ij}=0$ whenever $b_{ij}=0$. If $$ B = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & ...
1
vote
0answers
45 views

Principal eigenvector of non-negative symmetric block matrix is approximated by a linear combination of the principal eigenvectors of the blocks

Let $ M \in \mathbb{R}^{n \times n} = \begin{bmatrix} A & B \\ B^T & C \end{bmatrix} $ for some nonnegative $A \in \mathbb{R}^{k \times k}, B \in \mathbb{R}^{k \times n-k}, C \in \mathbb{R}^{n-...
15
votes
0answers
223 views

An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
13
votes
0answers
182 views

Is the set of power matrices decidable?

Let $\text{Mat}(n\times n,\mathbb{Z})$ denote the collection of integer $n\times n$ matrices. We say $M\in \text{Mat}(n\times n,\mathbb{Z})$ is a power matrix if there is an integer $k>1$ and a ...
1
vote
0answers
40 views

expected value of powers of a gaussian matrix

Let $Z$ be a fixed $d \times d$ matrix and let $G$ be a random $d \times d$ matrix with each entry i.i.d. $N(0, 1)$. Is it true that: $$\mathrm{Tr}(\mathbb{E}_G[ (Z^T + G^T)^\ell (Z + G)^{\ell-k-1}...
3
votes
1answer
91 views

Dimension of hermitian rank at most $k$ matrices over quaternions

In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
2
votes
0answers
107 views

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 ...
2
votes
2answers
137 views

Subspaces of real $n \times n$ matrices of dimension $O(n)$ [closed]

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 \...
9
votes
2answers
539 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})...
0
votes
0answers
25 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+\...