Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
1
vote
0answers
39 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
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...
0
votes
0answers
12 views
Intersection of parameterized cosets in a free-abelian group
Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set
\begin{equation}
S_{\{1,2\}} := \{...
2
votes
1answer
73 views
volume of parallelotope in $L^2(\mathbb R).$
Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product.
Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g.,
$$\{ f(...
22
votes
1answer
429 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
25 views
Invertible matrices T that commutate with the Jordan matrices A of size n [on hold]
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
166 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
52 views
Linear transformation where basis vectors are eigenvectors must be diagonal.Why? [on hold]
Right now I'm going through 3blue1brown's video on eigenvectors and I've been struggling for a while now at the very end. I didn't see any discussions about this particular part in the comment section,...
3
votes
0answers
118 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
55 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 ...
2
votes
0answers
62 views
How to find the best similarity transformation between two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$?
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 ...
-2
votes
0answers
25 views
Map a linear scale to a fixed [0-100] scale on a logarithmic basis [closed]
I have a linear scale [100-1000] and i want to map it on a [0-100] scale on a logarithmic basis.
100 becomes 0%
1000 becomes 100%
450 does not become 50% but a bigger percentage since the new scale ...
3
votes
1answer
115 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
244 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
48 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
148 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
$$\boldsymbol{X}\cdot\boldsymbol{1}=\boldsymbol{r}, \boldsymbol{1}^T\boldsymbol{...
0
votes
0answers
61 views
Representation of symmetric group as Cremona transformations
Question from me and a colleague:
Given a matrix
\begin{equation}
U =
\begin{bmatrix}
U_{11} & U_{12} \\
U_{21} & U_{22}
\end{bmatrix}
\quad \text{with } U_{22} \neq 0,
\end{equation}
...
0
votes
1answer
26 views
tensor stability of block-positive matrices
Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)
$\langle \psi |...
2
votes
1answer
47 views
Lie-algebra-like relation for totally symmetric 4-tensors
There are many totally symmetric real 4-tensors, $T_{ijkl}$, which satisfy the relation
$$T_{ijmn}T_{mnkl} + T_{ikmn}T_{mnjl} + T_{ilmn}T_{mnjk} = c T_{ijkl}$$
with some constant $c$. By the way ...
4
votes
0answers
111 views
How to formulate supercommutativity in a characteristic free way?
I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...
6
votes
0answers
66 views
mean distance between subspaces
Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...
5
votes
1answer
276 views
$(AB)^+\approx B^+A^+$ for $B$ “fat” enough?
Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse.
Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
-4
votes
0answers
132 views
On the determinant $\det[(i^2+dj^2)(\frac{i^2+dj^2}p)]_{1\le i,j\le(p-1)/2}$ with $p$ an odd prime
Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. We define the determinant $D(d,p)$ by
$$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
26 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
74 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
253 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
50 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
357 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
$$
\...
4
votes
1answer
122 views
The minimum rank of a matrix with a given pattern of zeros
For real matrices $A=(a_{ij})$ and $B=(b_{ij})$ of the same size, I write $A\prec B$ if $a_{ij}=0$ whenever $b_{ij}=0$.
If
$$ B = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & ...
1
vote
0answers
44 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
218 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 ...
12
votes
0answers
180 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:
$$\mathrm{Tr}(\mathbb{E}_G[ (Z^T + G^T)^\ell (Z + G)^{\ell-k-1}...
0
votes
0answers
23 views
PCA with zero and high correlation in data [migrated]
How would the eigen values look like when we apply PCA to a dataset with zero correlation between variables and when there is very high correlation between variables
3
votes
1answer
91 views
Dimension of hermitian rank at most $k$ matrices over quaternions
In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
2
votes
0answers
106 views
Positive and trace-preserving transformations with a common fixed point of full rank
The following problem which has been on my mind for a while now arises from the realm of quantum information involving quantum channels with a common fixed point of full rank, as well as majorization ...
2
votes
2answers
135 views
Subspaces of real $n \times n$ matrices of dimension $O(n)$ [closed]
The set of real $n \times n$ matrices forms a vector space over the reals. Given any set $S$ of $n \times n$ matrices, there is a basis $S' \subseteq S$ of size at most $n^2$ such that any $x \in S \...
9
votes
2answers
535 views
Certain matrices of interesting determinant
Let $M_n$ be the $n\times n$ matrix with entries
$$\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+\...
14
votes
2answers
503 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 ...
1
vote
0answers
546 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 ...
2
votes
0answers
75 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$....
1
vote
0answers
32 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 ...
14
votes
1answer
313 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 ...
1
vote
1answer
138 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}...
7
votes
1answer
181 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 ...
1
vote
0answers
39 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 ...
2
votes
0answers
61 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 (...
