Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
6,070 questions
8
votes
2
answers
91
views
Fast matrix-vector multiplication for a fixed 0-1 matrix
I have a fixed $n \times n$ matrix $M$ whose entries are all either 0 or 1. I want to compute the product $Mv$ for various vectors $v \in \mathbb{R}^n$ (or over other fields/rings).
Since $M$ is fixed ...
- 170
5
votes
1
answer
107
views
Element in conjugacy class of $\operatorname{SL}_2(\mathbb{Z})$ minimizing the lower left entry in absolute value
Let $G = \operatorname{SL}_2(\mathbb{Z})$. Consider an element $A \in G$ of trace $t$ of absolute value exceeding $2$, say. Let $C(A)$ denote the conjugacy class of $A$ in $G$. Let $c(A)$ denote the ...
- 30.7k
2
votes
0
answers
161
views
Subsets of $\operatorname{SL}_2(\mathbb{Z})$ closed under inverse, conjugation, and transpose
In the group $G = \operatorname{SL}_2(\mathbb{Z})$, there are several basic operations, namely inverse, transpose, and conjugation. Certainly the whole group $G$ is closed under these operations, and ...
- 30.7k
2
votes
0
answers
121
views
Decomposition $PLUP^{−1}$
This is a cross-post as I didn't get any answer. Let $A\in\mathrm{M}_n(\mathbb{C})$. If the leading principal minors (namely the determinants of the top left submatrices) of $A$ are non-zero, then ...
- 605
7
votes
1
answer
279
views
Decomposing elements in $\operatorname{SL}_2(\mathbb{Z})$ as products of symmetric matrices
It is proved in the paper Symmetric completions and products of symmetric matrices that the group $\mathfrak{S}$ generated by symmetric matrices in $\operatorname{SL}_2(\mathbb{Z})$ is not the whole ...
- 30.7k
0
votes
0
answers
134
views
Counterexample request: kernel of coproduct is not coproduct of kernels
Let $\mathcal{A}$ be an abelian category. Let $\{A_\alpha\xrightarrow{f_\alpha}B_\alpha\}$ be an infinite family of maps. Then there is an induced map:
$$\bigoplus_\alpha A_\alpha\xrightarrow{f=\...
- 2,413
6
votes
0
answers
286
views
Invertible perturbations of matrices
Let $P \in M_n(\mathbb{C})$ be a rank $k$ orthogonal projection, $k \geq 2$, and let $A_1, \ldots, A_r \in M_n(\mathbb{C})$ be matrices. Suppose that for every rank $k - 1$ orthogonal projection $Q &...
- 44.3k
0
votes
0
answers
93
views
Fourier transform and acoustic tensor
This is a repost from math stack since I have not received any answer
Let $C$ the fourth-order tensor of elastic constants which can be seen as as a linear transformation from Sym into Sym (matrices). ...
2
votes
0
answers
58
views
q-Deformed Laplacian of a graph
Suppose $G$ is a finite graph.
Let $A$ denote the adjacency matrix and $D$ the diagonal matrix whose entries are the degrees of vertices in $G$.
The matrix
$$
L_q = (1 - q^2)I - q A + q^2 D
$$
can be ...
- 2,215
2
votes
1
answer
125
views
Convergence of the correlation matrix for a specific type of random matrix product
Let $X= BM$ where
$B$ is a random $n\times m$ matrix with independent elements uniformly distributed on $[a, b]$.
$M$ is random $m\times m$ matrix with independent elements uniformly distributed on $[...
- 3,180
1
vote
0
answers
176
views
Representing tensors as algebraic sets
This is a question about elementary algebraic geometry on the edges with classical analytic geometry. So here, by an algebraic variety I mean just a solution set of a family of algebraic equations. ...
- 813
3
votes
0
answers
98
views
(Weak) Jordan-Chevalley decompositions over non-perfect fields
This is a reference/literature request.
Given a field $K$ and an endomorphism $x \colon V \to V$ of a finite-dimensional $K$-vector space it is well-known that the Jordan-Chevalley decomposition of ...
- 31
2
votes
0
answers
134
views
Linear-algebraic proof of $p$-group fixed-point theorem
Let $P$ be a $p$-group that acts on a finite set $X$. Let $X^P$ denote the $P$-fixed points of $X$. Then:
$$|X|\equiv |X^P|\pmod p$$
There is an easy combinatorial proof of this: $X$ is a disjoint ...
- 2,413
11
votes
1
answer
591
views
How close are elements of $\operatorname{SL}_2(\mathbb{Z})$ to symmetric matrices?
This is motivated by my earlier question: Minimal number of variables needed to parametrize $\operatorname{SL}_2(\mathbb{Z})$
A key step in the paper by Leonid Vaserstein cited in the linked question ...
- 30.7k
4
votes
1
answer
232
views
Cayley transform - determinant/permanent faster computation
Let $A$ be a $0/1$ matrix in $\mathbb Z^{n\times n}$ such that $I+A$ is invertible $\bmod 3$. Consider $Q=(I-A)(I+A)^{-1}$.
Let $Det(M)$ and $Per(M)$ be determinant and permanent respectively of ...
- 73
32
votes
1
answer
938
views
How large can subspaces of $U \otimes V$ be that avoid any pure tensors?
The Question is simple, yet I have encountered it multiple times in my mathematical life without finding an obvious answer, so I've decided to post it here.
Say $U, V$ are real vector spaces of ...
- 1,276
9
votes
2
answers
822
views
Linear algebraic lemma in Weil II
The following linear algebraic lemma is used in Weil II to prove properties of $\tau$-real sheaves:
Lemma Let $A$ be a complex matrix and let $\overline A$ be its complex conjugate. Then the ...
- 4,742
2
votes
0
answers
67
views
Upper-bounded version of the Gale-Ryser theorem
The standard Gale-Ryser theorem is for the existence of a $(0,1)$-matrix given exact row sums $R = (r_1, \dots, r_K)$ and exact column sums $C = (c_1, \dots, c_M)$. What if we relax the column sums ...
2
votes
0
answers
89
views
Asymptotic of dimensions of subvarieties of linear spaces that are nearly norm-dense in the unit balls
This is a follow-up to this question. I will start with the problem statement first (here, $B_n$ denotes the closed Euclidean unit ball of $\mathbb{R}^n$):
Fix sufficiently small $\varepsilon > 0$....
- 5,085
-4
votes
1
answer
406
views
Linear independent prime numbers? [closed]
Given a prime $p$ Let $$\phi(p):=\sum_{p_i <p} v_{p_i}(p-1) e_i$$ where $e_i$ is the $i$-th standard basis vector, $v_p(n)$ is the valuation of $n$ for the prime $p$ and $p_i$ is the $i$-th prime ...
15
votes
1
answer
872
views
(AC) and existence of basis of $\{0,1\}^X$ for any set $X$
For any set $X$, let $\{0,1\}^X$ be the collection of all functions $f:X\to\{0,1\}$. We make it into a vector space over the field $\mathbb{F}_2$ by endowing it with pointwise addition modulo $2$ and ...
- 54.4k
3
votes
1
answer
165
views
Eigenvalue of Grassmann graphs?
The Grassmann graph $G=\operatorname{Gr}_q(n,d)$ has vertex set the collection of $d$-dimensional subspaces of $\mathbb{F}_q^n$ and two vertices are adjacent iff their intersection has dimension $d-1$....
- 551
1
vote
0
answers
107
views
Determinantal elimination for $f_i=x_i(y+t_i)-1$: is there an analogue for $f_i=x_i(y+t_i z+s_i)-1$?
Consider the polynomials
$$
f_i = x_i (y + t_i) - 1,
$$
where the variables are $x_i$ and $y$.
Experiments indicate that, if the coefficients $t_i$ are algebraically independent, then there exists a ...
7
votes
2
answers
356
views
Unimodality / Totally non-negative properties of the sequence of the number of non-zero minors of a matrix
Consider a field $\mathbb{K}$ and a matrix $A \in \mathbb{M}_n(\mathbb{K})$. Let's define for each $0\le k \le n$ , the number $N_k$ defined as the number of non-zero minors of size k of A. For k=0 , ...
- 73
3
votes
0
answers
99
views
Characterizing polyhedra via "finitely many faces"
I recently tried to see in how far polyhedral geometry can be reduced to the study of convex sets with finitely many faces. In other words, I tried to replace "finitely generated" by "...
- 14.5k
2
votes
0
answers
104
views
A Coxeter group associated to finite dimensional acyclic algebras
Let $A=KQ/I$ be an acyclic quiver algebra with Cartan matrix $U$ and let $n$ be the number of vertices of $Q$.
For example when $A=KP$ is the incidence algebra of a finite poset $P$, then $U$ is just ...
- 28.2k
3
votes
1
answer
143
views
Jordan subalgebra with equal enveloping algebra
Let $J_0\subset J$ be Jordan algebras of hermitian $n\times n$ matrices, and let $A \subset M_n(\mathbb C)$ be the *-algebra generated by $J$.
Suppose that $J\subset A$ is the universal embedding of $...
- 809
0
votes
2
answers
103
views
Symmetric matrix inversion if diagonalization in larger dimension is known
Given an $n \times n$ orthogonal matrix $U$ (i.e., $U^T U = \mathbb I_{n\times n}$), then for an $n \times n$ diagonal matrix $D$, it is easy to verify that $ \left( U^T D U \right)^{-1} = U^T D^{-1} ...
- 343
2
votes
0
answers
120
views
Is there a 'determinant' of a two-variable function when treated as a linear map?
A two variable real function $F(y,x)$, defined on $[c,d] \times [a,b]$, can be thought of as a linear map between functions of different domains by:
$$
g(y) = \int^a_bF(y,x)f(x)dx
$$
This has very ...
- 121
6
votes
1
answer
431
views
Direct sum and tensor product
Given a finite dimensional vector space $V$ over a characteristic zero field, is there any canonical isomorphism between $V^{\oplus \dim V}$ and $V^{\otimes 2}$?
Canonical means that does not depend ...
- 2,346
5
votes
0
answers
104
views
What does the automorphism group of a norm cone look like in general?
Let $(V, \lVert \cdot \rVert)$ be a finite-dimensional real normed space, and let $C \subseteq \mathbb R \oplus V$ be the norm cone of $V$; that is, $C$ consists of all $(t, v)$ for which $\lvert t \...
- 181
1
vote
0
answers
104
views
Characteristic polynomial of block tridiagonal matrix
Suppose that I have an $nk \times nk$ matrix of the form
$$
T_n = \left[\begin{array}{cccccc}
A&B&&&&\\
B^T&A&B&&&\\
&B^T&A&B&&\\
&&\...
- 13.7k
1
vote
1
answer
169
views
How to prove positive definiteness of a matrix under given premises?
${\bf A} \in {\Bbb R}^{n \times n}$ is a symmetric positive definite matrix, whose diagonal elements are all positive while off-diagonal elements are all non-positive. ${\bf U} \in {\Bbb R}^{n \times ...
- 75
0
votes
0
answers
98
views
Periodicity of Coxeter matrices related to Dynkin diagrams
Let $Q$ be a Dynkin quiver of Dynkin type $A_n$ for even $n$ or $D_n$ for $n$ odd.
$Q$ is called symmetric if the orientation is stable under the canonical involution of the Dynkin quiver (which is ...
- 28.2k
1
vote
0
answers
424
views
A question in matrix analysis and linear algebra related to Hermitian Toeplitz matrix
Consider a Hermitian Toeplitz matrix and modify it with entries in leading diagonal as $H_{n\times n}(x,x) = x^2 + \lambda,x=0,1,2,\dotsc n-1$. Now we choose $\lambda\in\mathbb{R}$ such that the ...
- 544
0
votes
0
answers
61
views
relation between the approximate $\gamma_2$ norms of Boolean matrix and sign matrix
For a real matrix $A$, its $\gamma_2$-norm is defined as
$$\gamma_2(A)=\min_{X,Y:A=XY}||X||_{row}||Y||_{col},$$
where $||\cdot||_{row},||\cdot||_{col}$ are defined as the maximum $\ell_2$-norm of row ...
- 551
2
votes
1
answer
208
views
A Loewner ordering problem
Let $A$ be a positive definite diagonal matrix, $B$ be a real matrix, and $C$ be a complex matrix. All are square matrices of dimension $n$. I am wondering if it's true that
$$\Big\|A+B^\top C^* C B\...
- 71
3
votes
1
answer
334
views
For which $k$ does a generic choice of $k$ $n \times n$ matrices span a subspace of $\mathrm{GL}(n)$?
If $\rho(n)$ are the Radon-Hurwitz numbers, then for $k \leq \rho(n)$ it is possible to find $k$ $n \times n$ real-valued matrices $A_1, \dots, A_k$ so that for any $(a_1,\dots,a_k) \neq 0$, the ...
- 507
1
vote
0
answers
144
views
Upper bound on the nullity of graph over the ring $\mathbb Z_N$
$\DeclareMathOperator{\nullity}{nullity}$Given a field $F$ and a simple undirected graph $G$ on $n$ vertices, let $S(F,G)$ denote the set of all symmetric $n\times n$ matrices $A$ with entries in $F$ ...
- 521
1
vote
0
answers
72
views
Dimension of Chowla subspaces
Definition (Chowla subspace).
Let $K \subseteq L$ be a field extension and let $A$ be a $K$-subspace of $L$.
We say that $A$ is a Chowla subspace if for every $a \in A \setminus \{0\}$ one has
$$[K(a):...
- 379
8
votes
2
answers
403
views
Vandermonde-type closed form for products of $3\times3$ minors?
Question. The classical Vandermonde identity says
$$
\prod_{1\le i<j\le n}(t_i-t_j)=
\det\!\begin{pmatrix}
1 & \cdots & 1\\
t_1 & \cdots & t_n\\
\vdots & \ddots & \vdots\\
...
-6
votes
1
answer
122
views
Determining if binary matrix with specific form has full rank [closed]
I have the following 15x15 binary matrix with a specific form:
$$\begin{bmatrix}
1&1&0&0&0&0&1&1&1&1&1&1&0&0&0 \\
1&0&1&0&0&...
- 1
2
votes
0
answers
224
views
Gauss Sums over Conjugacy classes of $M_n(F_q)$
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\Tr{Tr}$Let $V := M_n(F_q)$ be $n \times n$ matrices over a finite field $F_q$. Let $X$ be a conjugacy class in $V$ whose characteristic ...
- 611
0
votes
0
answers
60
views
analyzing the sensitivity of two matrix expressions
I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other.
We are given:
$$\...
- 11
1
vote
0
answers
155
views
Diagonalization of a monomial matrix with constraints
Definitions
Let $G$ be a finite subgroup of $U(n)$ and let $\mathcal{D} \subset U(n)$ denote the group of $n\times n$ diagonal, unitary matrices. We'll say that a matrix $T$ is diagonalizable over $G$ ...
- 123
7
votes
0
answers
626
views
Linear independence of composition of polynomials
Let $k, d \in \mathbb{Z}_{\geq 1}$ be positive integers.
I want to prove the following.There exists a large enough $r \in \mathbb{Z}_{\geq 1}$ (depending on $k$ and $d$) and a Zariski open set $U \...
- 409
20
votes
1
answer
800
views
Dimension of the variety of commuting matrices
Let C(m,n) be the variety of of m-tuples of commuting matrices of size n. What is the dimension of this variety? This was an open problem in Guralnick's 1992 paper "A note on commuting pairs of ...
- 311
1
vote
1
answer
157
views
Graph-theoretic conditions leading to singularity of adjacency matrices
Let $G$ be a directed graph on $n$ vertices, with adjacency matrix $A \in \{0,1\}^{n \times n}$ (loops allowed). It is well-known that if one row of $A$ is a real linear combination of other rows, ...
- 4,150
-1
votes
2
answers
118
views
Constructing an orthonormal set with given projections in a direct sum decomposition
Let $V$ be an $n$-dimensional real inner product space. Suppose we have $k\leq n/2$ orthonormal vectors $u_1, u_2, \dots, u_k \in V$.
Assume that there exist pairwise orthogonal subspaces $A,B,C \...
- 11
19
votes
2
answers
1k
views
Off-diagonal entries of $(AA^T)^{1/2}$ for the bipartite adjacency matrix of a tree
Let $T$ be a tree with a bipartition of its vertices into sets $X = \{v_1, \dots, v_m\}$ and $Y = \{w_1, \dots, w_n\}$. Define the $m \times n$ bipartite adjacency matrix $A$ by
$$ A_{ij}=\begin{cases}...
- 4,503