Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
6,058 questions
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Covering lattice points with planes and lines
The following question was asked on a Chinese contest:
Prove that there exists a real constant $c>0$ such that if all lattice points inside and on the boundary of a convex polyhedron in $ \mathbb{...
- 3,143
26
votes
1
answer
747
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 ...
- 1,196
8
votes
2
answers
726
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 ...
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votes
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answers
37
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Levinson Recursion With Sub-Singular Hermitian Toeplitz Matrices Fails for Complex Inputs [closed]
I’m implementing a version of the Levinson recursion that should handle sub-singular Hermitian Toeplitz matrices. My code works perfectly when the Toeplitz entries are real, but it fails as soon as ...
2
votes
0
answers
65
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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
80
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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
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votes
1
answer
391
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 ...
- 3,761
15
votes
1
answer
843
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(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 ...
3
votes
1
answer
160
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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
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vote
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103
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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
350
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 , ...
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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 "...
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102
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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 ...
- 28.1k
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answer
142
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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 $...
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2
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102
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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
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answers
120
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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 ...
6
votes
1
answer
427
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 ...
- 2,336
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102
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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 \...
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98
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Characteristic polynomial of block tridiagonal matrix
Suppose that I have an $nk \times nk$ matrix of the form
$$
T_n = \left[\begin{array}{cccccc}
A&B&&&&\\
B^T&A&B&&&\\
&B^T&A&B&&\\
&&\...
- 13.7k
1
vote
1
answer
153
views
How to prove positive definiteness of a matrix under given premises?
${\bf A} \in {\Bbb R}^{n \times n}$ is a symmetric positive definite matrix, whose diagonal elements are all positive while off-diagonal elements are all non-positive. ${\bf U} \in {\Bbb R}^{n \times ...
- 75
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0
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97
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Periodicity of Coxeter matrices related to Dynkin diagrams
Let $Q$ be a Dynkin quiver of Dynkin type $A_n$ for even $n$ or $D_n$ for $n$ odd.
$Q$ is called symmetric if the orientation is stable under the canonical involution of the Dynkin quiver (which is ...
- 28.1k
1
vote
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423
views
A question in matrix analysis and linear algebra related to Hermitian Toeplitz matrix
Consider a Hermitian Toeplitz matrix and modify it with entries in leading diagonal as $H_{n\times n}(x,x) = x^2 + \lambda,x=0,1,2,\dotsc n-1$. Now we choose $\lambda\in\mathbb{R}$ such that the ...
- 544
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58
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relation between the approximate $\gamma_2$ norms of Boolean matrix and sign matrix
For a real matrix $A$, its $\gamma_2$-norm is defined as
$$\gamma_2(A)=\min_{X,Y:A=XY}||X||_{row}||Y||_{col},$$
where $||\cdot||_{row},||\cdot||_{col}$ are defined as the maximum $\ell_2$-norm of row ...
- 551
2
votes
1
answer
203
views
A Loewner ordering problem
Let $A$ be a positive definite diagonal matrix, $B$ be a real matrix, and $C$ be a complex matrix. All are square matrices of dimension $n$. I am wondering if it's true that
$$\Big\|A+B^\top C^* C B\...
- 71
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votes
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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
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142
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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
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vote
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71
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
$$[K(a):...
- 379
8
votes
2
answers
401
views
Vandermonde-type closed form for products of $3\times3$ minors?
Question. The classical Vandermonde identity says
$$
\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
121
views
Determining if binary matrix with specific form has full rank [closed]
I have the following 15x15 binary matrix with a specific form:
$$\begin{bmatrix}
1&1&0&0&0&0&1&1&1&1&1&1&0&0&0 \\
1&0&1&0&0&...
- 1
2
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0
answers
224
views
Gauss Sums over Conjugacy classes of $M_n(F_q)$
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\Tr{Tr}$Let $V := M_n(F_q)$ be $n \times n$ matrices over a finite field $F_q$. Let $X$ be a conjugacy class in $V$ whose characteristic ...
- 611
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0
answers
60
views
analyzing the sensitivity of two matrix expressions
I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other.
We are given:
$$\...
- 11
1
vote
0
answers
154
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
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votes
0
answers
618
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
798
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
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vote
1
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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
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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
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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
$$ A_{ij}=\begin{cases}...
- 4,503
42
votes
14
answers
4k
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Alternative proofs that two bases of a vector space have the same size
Let's stick to finite-dimensional vector spaces. I'm teaching linear algebra right now, and soon I'll have to prove that any two bases of a finite-dimensional vector space have the same size. I've ...
2
votes
0
answers
76
views
Volume of projections of a $k$-cube into coordinate subspaces
Let $\{v_i\}_{i=1,\ldots, k}$ be $k$ orthonormal vectors in $\mathbb{R}^n$, with $1 \le k < n$, and consider the $k$-cube
$$
V := \Big\{\sum_i t_iv_i \mid 0 \le t_i \le 1 \text{ for all } 1 \le i \...
- 438
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views
Can we construct a polynomial $T$ such that $g(\alpha)$ is a root of $T$, for all $\alpha \in Z(f)$, where $g, f$ are two monic polynomial?
Let there be two monic polynomials $f(x), g(x) \in \mathbb{Z}/(p^k \mathbb{Z})[x]$, where $p$ is an odd prime number and $k \geq 2$. Can we construct a polynomial $T$ such that $g(\alpha)$ is a root ...
- 311
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votes
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121
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A higher dimensional version of the Bruhat decomposition
The classical Bruhat decomposition states that every invertible matrix $M$ over a field $K$ can be written as $M= U_1 P U_2$ with $U_i$ upper triangular matrices and $P$ a permutation matrix. We can ...
- 28.1k
1
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76
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Finding the generalized inverse of $A_{n \times n}$ recursively [closed]
Suppose I have a matrix $A_{n \times n} , n \ge 2$. Let $n_1,n_2,\cdots,n_k$ be natural numbers such that $\sum_{i=1}^k n_i = n.$ Define $n_i^* = n_1+\cdots+n_i$ Then , $A$ can be partitioned as :
$\...
- 131
7
votes
1
answer
441
views
The multiplication table of a semigroup as a matrix
I'm not very familiar with the theory of semigroups, so sorry if this is well-known or easy.
Let $G=\{x_1,...,x_n\}$ be a finite semigroup with $n$ elements.
Let $T$ be a field and $K=T(y_1,...,y_n)$ ...
- 28.1k
3
votes
0
answers
103
views
A parameter measures the number of entries close to zero?
Let $A\in\mathbb{R}^{n\times n}$. Let $s(A)$ be the number of entries of $A$ that are in $(-1,0)\cup(0, 1)$.
I am wondering is there any criteria (maybe some norms etc） implies that the size of $s(A)$ ...
- 551
3
votes
1
answer
259
views
Why is this function multiplicative over the monoid of $3\times 3$-circulant matrices that sum of rows is zero
Let $F$ be a field whose characteristic does not divide $6$.
$\DeclareMathOperator{\circ}{circ} \DeclareMathOperator{\had}{had} \newcommand{\Circ}{\mathrm{Circ}}$
$\circ(a,b,c)$ is the circular matrix ...
- 501
3
votes
0
answers
233
views
If resultant $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ have a nontrivial factors then can $f(x)$ also have a nontrivial factors?
This is a follow-up question to this question. In that question, we learned that if, $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$.
Now, my question is:
If $T(y) = ...
- 311
9
votes
0
answers
562
views
Is it hard to decide if two codes have the same weight enumerator polynomial?
Consider the following decision problem, which we will call COMPARE. We are given as input a pair $(V_0, V_1)$ of linear codes in $\mathbb{F}_2^n$, and asked to decide whether $V_0, V_1$ have the same ...
- 401
0
votes
0
answers
127
views
All possible images of a vector after averaging
For a given $x \in \mathbb{R}^n$, is there a name given for the set of all possible $y\in \mathbb{R}^n$ that can be obtained by iteratively selecting a subset of entries $S\in \{1,\dots,n\}$ and ...
- 4,131
3
votes
1
answer
318
views
Generalization of a well-known linear algebraic result to modules
There is a well-known result in linear algebra that is stated below:
Suppose that $g, f_{1}, f_{2}, \ldots, f_{r}$ are linear functionals on a vector space $V$ and let $N, N_{1}, N_{2}, \ldots, N_{r}$ ...
- 33
9
votes
5
answers
1k
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Does an irreducible representation $\;p:G\rightarrow (V \rightarrow V)$ always span the whole space of maps $V\rightarrow V$?
EDIT(Andy Putman): Since it's written in what I think is a confusing way, I'm going to rewrite the question in a different language. The original question is below.
Let $G$ be a group and let $V$ be ...
- 269