Questions tagged [linear-algebra]

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

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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 ...
Ayna's user avatar
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8 votes
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 ...
Fumiyo Eda's user avatar
8 votes
3 answers
808 views

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 ...
Pace Nielsen's user avatar
8 votes
1 answer
2k views

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 ...
Aaron Meyerowitz's user avatar
8 votes
2 answers
2k views

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 ...
Atnap's user avatar
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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-...
Jeff's user avatar
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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 (...
Per Alexandersson's user avatar
8 votes
1 answer
1k views

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 ...
Gunnar Þór Magnússon's user avatar
8 votes
1 answer
1k views

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} ...
jschn's user avatar
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8 votes
3 answers
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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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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 ...
Mark Bell's user avatar
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2 answers
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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 ...
Favst's user avatar
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1 answer
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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 ...
Igor Rivin's user avatar
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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 $S$ ...
Mikhail Goltvanitsa's user avatar
7 votes
1 answer
354 views

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)=\...
Bill Bradley's user avatar
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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 $Tr[B^s (B R^2 B)^t] \ge Tr[B^s (R B^2 R)^t]$ ?
jjcale's user avatar
  • 2,768
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 ...
Federico Poloni's user avatar
7 votes
3 answers
1k views

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 ...
J. Doi's user avatar
  • 71
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3 answers
688 views

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 $\...
Jiro's user avatar
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2 answers
583 views

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 $D$ ...
Jiro's user avatar
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0 answers
889 views

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 }...
Mats Granvik's user avatar
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7 votes
3 answers
2k views

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?
heiko's user avatar
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0 answers
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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 ...
Mare's user avatar
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7 votes
1 answer
596 views

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}(...
ghc1997's user avatar
  • 763
7 votes
2 answers
573 views

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{\...
Vít Tuček's user avatar
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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. ...
SIB's user avatar
  • 241
7 votes
2 answers
1k views

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 ...
Theo Johnson-Freyd's user avatar
7 votes
2 answers
1k views

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 $M_n$. ...
Chris Heunen's user avatar
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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 ...
user avatar
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 ...
A.B.'s user avatar
  • 407
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 ...
Mark Wildon's user avatar
  • 10.8k
7 votes
2 answers
914 views

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}...
Analysis Student's user avatar
7 votes
1 answer
499 views

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\...
H A Helfgott's user avatar
  • 19.3k
7 votes
3 answers
3k views

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 ...
Nate Eldredge's user avatar
7 votes
1 answer
491 views

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 ...
Jules's user avatar
  • 463
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}^{...
Jochen Glueck's user avatar
7 votes
4 answers
2k views

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 ...
Jiahao Chen's user avatar
  • 1,870
6 votes
1 answer
209 views

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 ...
Iosif Pinelis's user avatar
6 votes
2 answers
1k views

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/...
Robert Bailey's user avatar
6 votes
1 answer
478 views

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 ...
T. Amdeberhan's user avatar
6 votes
1 answer
948 views

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}\...
Paul Christiano's user avatar
6 votes
3 answers
7k views

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 ...
Ian Martin's user avatar
6 votes
1 answer
143 views

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 ...
Matteo Cati's user avatar
6 votes
1 answer
324 views

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\...
driss-alamilouati's user avatar
6 votes
1 answer
928 views

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 $...
anderstood's user avatar
6 votes
0 answers
216 views

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 \...
user avatar
6 votes
0 answers
337 views

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{...
solver6's user avatar
  • 291
6 votes
1 answer
594 views

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 ...
wlad's user avatar
  • 4,823
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) ...
Kenneth's user avatar
  • 63
6 votes
2 answers
2k views

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 ...
Suvrit's user avatar
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