Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
588
questions
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A question on eigenvalues
Let $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ be linearly independent Hermitian matrices in the the space of $6$ by $6$ Hermitian matrices as a vector space over $\mathbb{R}$. Does there always ...
8
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5
answers
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Eigenvalues of A+B where A is symmetric positive definite and B is diagonal
If I have a symmetric positive definite matrix A and a diagonal matrix B, and I know the eigenvalues of both A and B (by iterative numerical computation in A's case and trivially for B), is there any ...
8
votes
3
answers
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Does a left basis imply a right basis, without AC?
If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$?
(The original question appears below. But this shorter question gets at the ...
8
votes
1
answer
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Complexity of finding a 0-1 vector in a subspace or showing that there is none
This question, is a slightly different disguise (see below), came up in discussions of this question about equitable partitions
A $0,1$ vector in $\mathbb{Z}^n$ is any vector with all entries $0$ and ...
8
votes
2
answers
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Expectation of trace of nth power of unitary matrices
I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
8
votes
1
answer
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Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?
How can I efficiently compute
$\mathrm{trace}(A(B^{-1}))$
where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of non-...
8
votes
1
answer
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Integer solution to special system of linear equations
This problem appear in my research, but I am unable to solve it.
There should be an easy argument, but I have not yet found it.
Informal version
An integer $k\geq 2$ is fixed.
We are given a matrix (...
8
votes
1
answer
1k
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Calculating a curvature tensor by polarization
I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
8
votes
1
answer
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Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
8
votes
3
answers
460
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Sets which are not fixed by any non-identity isomorphism
Consider a finite dimensional vector space $V$ over a field (finite or infinite but big enough). I am looking for a subset $W$ of $V$ such that for any bijective but non-identity linear map $T: V \...
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3
answers
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Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?
Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.
Does an upper ...
8
votes
2
answers
2k
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Almost orthogonal vectors in subsets of euclidean space
Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of almost-orthogonal sets ...
8
votes
1
answer
491
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Casson invariant
Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does ...
8
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2
answers
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Polynomial roots in the ring extension
Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
7
votes
1
answer
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Injectivity of matrix "fingerprint"
Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries.
For any matrix $A$, define
$$ D_A(i,j)=\...
7
votes
1
answer
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A conjectured trace inequality for some products of powers of matrices
Let $B, R\in M_{n}(\mathbb{C})$ hermitian and $B$ positive semidefinite.
Let $s,t \in \mathbb{R}$ and $s,t \ge 0$ .
Does then hold $Tr[B^s (B R^2 B)^t] \ge Tr[B^s (R B^2 R)^t]$ ?
7
votes
2
answers
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Factorizing a block symmetric matrix
Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible.
I would like to ...
7
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3
answers
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How to calculate inverse of sum of two Kronecker products with specific form efficiently?
I have a matrix with specific form of $A\otimes I + B\otimes J$ where $A$ and $B$ are general dense matrices, $n\times n$. $I$ is an $m\times m$ identity matrix. $J$ is a $m \times m$ dense matrix ...
7
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3
answers
688
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Generalized Characteristic Polynomial with Unimodular Roots
Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{m_1}, \dots, z^{m_N})$ with $z\in\mathbb{C}$ and positive integers $m_1, \dots, m_N$.
The generalized characteristic polynomial of a matrix $\...
7
votes
2
answers
583
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Minimize spectral norm under diagonal similarity
Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e.,
$$
s(A) = \min_{D} \lVert D^{-1} A D\rVert_2,
$$
where $D$ ...
7
votes
0
answers
889
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The Möbius function as eigenvalues
Let the $N$ by $N$ matrix $A$ be defined by the tetration:
$$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
7
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3
answers
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Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
7
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0
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337
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A homological algebra approach to the Union-closed sets conjecture
I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
7
votes
1
answer
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Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
7
votes
2
answers
573
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Laplace-like / cofactor expansion for Pfaffian
Wikipedia presents a recursive definition of the Pfaffian of a skew-symmetric matrix as $$ \operatorname{pf}(A)=\sum_{{j=1}\atop{j\neq i}}^{2n}(-1)^{i+j+1+\theta(i-j)}a_{ij}\operatorname{pf}(A_{\hat{\...
7
votes
4
answers
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Positive solutions of linear Diophantine equations
Let $A$ be a non-negative integer $k\times n$-matrix (i.e. each entry is non-negative and integer) with $rank(A) = k < n$. Let $b$ be a $k$-dimensional vector with positive integer entries. ...
7
votes
2
answers
1k
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Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
7
votes
2
answers
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When is this map completely positive?
Consider the complex $n$-by-$n$ matrices $M_n$.
Suppose that $A_i$, for $i=1,\ldots,n^2$, satisfy $\mathrm{Tr}(A_i^*
A_j)=\delta_{ij}$, so that together they form an orthonormal basis for
$M_n$. ...
7
votes
1
answer
870
views
Does this cross-product norm inequality hold?
I asked this on MSE over a month ago, but the one answer I got doesn't seem to work.
Let $\times$ denote the cross-product. $\;$ Is it the case that
For all unit vectors $\:\mathbf{x}\hspace{.01 ...
7
votes
3
answers
293
views
Sets from $(F_2)^n$ which are not fixed by any non-identity isomorphism
This is a followup question to the discussion in the comments of
Sets which are not fixed by any non-identity isomorphism
So consider a finite $n$-dimensional vector space $V$ over $F_2$. For which ...
7
votes
1
answer
436
views
Hopfian modules
My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
7
votes
2
answers
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Principal ideal ring, does there exist an invertible matrix such that certain matrix is upper triangular?
I asked here on Math Stack Exchange the following question.
Let $R$ be a principal ideal ring. If $A$ is any $p \times q$ matrix over $R$, then does there exist an invertible matrix $U$ in $\text{M}...
7
votes
1
answer
499
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Counting points on the intersection of a box and a lattice
Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\...
7
votes
3
answers
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Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space
I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask.
First, consider the following form ...
7
votes
1
answer
491
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Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently
What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$?
Do we have to calculate $A^{-1}b$, or is this not necessary?
edit: I forgot to mention that A is symmetric and ...
7
votes
3
answers
552
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
7
votes
4
answers
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Is there a name for the matrix equation A X B + B X A + C X C = D?
I happen to be working on a problem that reduces to solving the following equation:
$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$
where A through D are known matrices ( A, B, D ...
6
votes
1
answer
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An inequality for certain positive-definite matrices
Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with ...
6
votes
2
answers
1k
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Systems of simultaneous real quadratic equations
Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/...
6
votes
1
answer
478
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A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
6
votes
1
answer
948
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Bounding the second derivative of the log-determinant
I'm trying to use the log-determinant to regularize an optimization problem. To make the argument work, I need to bound the second derivative of the log-determinant.
I need to prove that $\text{Tr}\...
6
votes
3
answers
7k
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Calculating the Perron-Frobenius eigenvector of a positive matrix from limited information
In the background of this question is a matrix $A$, all of whose elements are positive. The Perron-Frobenius theorem tells us that the eigenvalue with largest absolute value is real, and that there ...
6
votes
1
answer
143
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How to construct a skew Hadamard matrix of order 756?
Where can I find the construction for a skew Hadamard matrix of order 756?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard matrices and Seberry - On skew Hadamard ...
6
votes
1
answer
324
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construction of matrices verifying an identity
Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that B is nonsingular and $AB\neq BA$. Can we always find real numbers $t_1,⋯,t_p$ such that
$$B\left(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\right)A=A\...
6
votes
1
answer
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Proving that the kernel of this matrix is of dimension 2
(Edit : see at the bottom of the question for an additional surprising possible hint.)
Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when $...
6
votes
0
answers
216
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Is this function embeddable in Euclidean space?
Let $X = \{v_1,\ldots,v_n\}$ be a set of vectors non-zero vectors $v_i \ge 0$ and such that the vectors are pairwise linear independent. Define a function on this set $X$:
$$d(v,w) = 1-\frac{2 \...
6
votes
0
answers
337
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Asymptotically nilpotent Lie sets of matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$.
Question 1. Assume that $\mathcal{A}$ is the subset of $\textbf{...
6
votes
1
answer
594
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Is there a conceptual reason why every square complex matrix is similar to a complex-symmetric matrix?
The question is maybe a bit vague, but like the title says: Every square complex-matrix $M$ is equal to $P S P^{-1}$ where $S = S^T$. The proof begins by taking the Jordan Normal Form of $M$, and then ...
6
votes
4
answers
642
views
Reference for an algebraic group preserving a cubic form
Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...
6
votes
2
answers
2k
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Tight bound for sum of entries of the inverse of a nonnegative matrix
While playing around with certain non-negative matrices, I got stuck at the following question.
Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...