All Questions
414 questions
17
votes
2
answers
1k
views
The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
- 43.2k
17
votes
1
answer
3k
views
2x2 subdeterminants of a matrix
If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...
- 1,309
17
votes
1
answer
7k
views
How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?
I asked this question in Math Stack Exchange earlier here: https://math.stackexchange.com/questions/1199380/what-is-the-intuition-behind-how-can-we-interpret-the-eigenvalues-and-eigenvec and since I ...
- 271
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
17
votes
0
answers
704
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
16
votes
2
answers
2k
views
How to prove the determinant of a Hilbert-like matrix with parameter is non-zero
Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\...
- 163
16
votes
7
answers
2k
views
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the ...
- 355
16
votes
6
answers
13k
views
Showing block diagonal structure of matrix by reordering
Suppose we have a block-diagonal matrix $M$, but the block diagonal structure is not immediately apparent from looking at the matrix because the rows/columns are shuffled.
I wish to find a reordering ...
- 911
16
votes
1
answer
2k
views
Overlapping Gershgorin disks
We all know Gershgorin's Circle Theorem, which I will summarise for convenience. Let $A=(a_{ij})$ be an $n\times n$ complex matrix. Define the disks $D_1,\ldots,D_n$ by $$D_i = \Bigl\{ z : |z-a_{ii}|\...
- 37.7k
16
votes
2
answers
4k
views
The singular values of the Hilbert matrix
The $n\times n$ Hilbert matrix $H$ is defined as follows
$$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$$
What is known about the singular values $\sigma_1 \geq \cdots \geq \sigma_n$ of $H$?
...
- 3,217
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 163
15
votes
1
answer
1k
views
Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$
Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, ...
- 19.6k
15
votes
1
answer
2k
views
Which finite groups have no irreducible representations other than characters?
A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
- 11.3k
15
votes
2
answers
4k
views
Comparison of the norm of a matrix and its entry-wise absolute value.
It is an easy fact that for a matrix $A \in M_n(\mathbb C)$, the matrix $A' = (|A(i,j)|)_{i,j \leq n}$ has a larger operator norm than $A$. By operator norm I mean the norm as an operator on $\ell^2_n$...
- 9,486
15
votes
2
answers
7k
views
Efficient rank-two updates of an eigenvalue decomposition (or more generally SVD)
Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. Golub, et al.1 and Bunch, et al.2 have shown that given such an $A$, the eigenvalue decomposition of $A+\rho xx^t$ may be computed ...
- 947
15
votes
3
answers
6k
views
Concavity of $\det^{1/n}$ over $HPD_n$.
One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...
15
votes
4
answers
7k
views
Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
- 153
15
votes
1
answer
2k
views
Necessary and sufficient conditions for a sum of idempotents to be idempotent
Given: a finite list of $n$-by-$n$ idempotent complex matrices $E_1, E_2, \ldots, E_k$.
If all pairwise products $E_i E_j$ (with $i \neq j$) are zero, it is trivial to show the sum $E_1 + E_2 + \cdots ...
- 151
15
votes
7
answers
6k
views
Binary matrices with constant row and column sums
My question is about $m \times n$ binary matrices (aka $\{0,1\}$-matrices), whose rows all sum to the same value, and whose columns all sum to the same value (but these two values may be different).
...
- 878
15
votes
1
answer
1k
views
Existence of double eigenvalue
Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent.
Must there exist a nonzero real linear combination $aA + bB$ which has a repeated ...
- 42.8k
14
votes
1
answer
581
views
How flexible is the infinite-dimensional torus?
Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group.
Problem 1. Is it true that for ...
- 41.8k
14
votes
3
answers
3k
views
Diagonalizing a Certain Real and Symmetric Toeplitz Matrix
Consider $0\leq \alpha\leq 1$, and let $A_{\alpha}$ be the Toeplitz $n\times n$ matrix given by
$$
A_\alpha := \begin{bmatrix}
1 & \alpha & \alpha^2 & \ldots &\alpha^{n-1} \\\
\alpha ...
- 3,626
14
votes
3
answers
1k
views
The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their eigenvalues, what about pairs?
For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of ...
user76098
13
votes
2
answers
414
views
Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
- 23.3k
13
votes
2
answers
6k
views
Parametrization of positive semidefinite matrices
We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...
- 313
13
votes
2
answers
913
views
Almost Hadamard matrices
As well-known, a Hadamard matrix is a square matrix with all coefficients $\pm 1$
and pairwise orthogonal rows or columns. Such matrices exist conjecturally
in every dimension divisible by $4$. Call ...
- 17.9k
13
votes
3
answers
746
views
Is there a row vector $x$ with integer entries such that no entry of $xM$ is $0 \text{ (mod }p\text{)}$?
Let $p$ be a prime and let $M$ be an $n \times m$ matrix with integer entries such that $M\vec{v} \not\equiv \vec{0} \text{ (mod }p\text{)}$ for any column vector $\vec{v} \neq \vec{0}$ whose entries ...
- 305
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
12
votes
3
answers
4k
views
Status of Hadamard matrix conjecture
I would like to know if any progress has been made on Hadamard conjecture :
Hadamard matrix of order $4k$ exists for every positive integer $k$.
- 121
12
votes
1
answer
3k
views
Matrix inversion lemma with pseudoinverses
The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own.
Suppose we pick $n$ values $x_1,\...
- 28.6k
12
votes
1
answer
267
views
Total positivity of $q$-Pascal matrix?
A matrix of real numbers is called totally positive if all its minors are non-negative. A well-known example is the Pascal matrix $(\binom{i}{j})$.
Is it true that the minors of the $q$-Pascal matrix ...
- 5,622
12
votes
0
answers
218
views
Which ordering of factors is needed to obtain this kind of determinantal inequalities?
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
- 13.4k
12
votes
2
answers
2k
views
Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...
- 121
12
votes
0
answers
825
views
Eigenvalues of permutations of a real matrix: how complex can they be?
This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
- 13.4k
12
votes
1
answer
404
views
Uses of Volodin's construction of algebraic K-theory
There is a construction of the algebraic K-theory groups $K_i(R)$ of a ring $R$ by Volodin. He gave an explicit construction of the plus-construction $BGL(R)^+$ as the quotient of the bar construction ...
- 637
11
votes
1
answer
328
views
$2$-adic valuation of Schur $P$-functions in the power-sum basis
For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...
- 3,446
11
votes
2
answers
2k
views
Inverse of a small submatrix
Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
- 241
11
votes
1
answer
467
views
Correspondence between matrix multiplication and a graph operation of Lovász
In his book "Large networks and graph limits", Lovász describes a multiplication operation (he calls it concatenation) on "bi-labeled graphs". An $(m,n)$ bi-labeled graph is a ...
- 2,339
11
votes
0
answers
248
views
Is my matrix perturbation analysis legitimate?
I am not a matrix theorist, or numerical linear algebra expert, but I have a problem and my proposed solution leads me to a question that I cannot answer.
I can give more details, but the gist is ...
- 12.7k
11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
- 43.2k
11
votes
1
answer
453
views
A variant of Cholesky decomposition involving binary matrices
Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
- 3,463
11
votes
5
answers
3k
views
Strassen Algorithm 7 multiplications
Strassen Algoritm is a well-known matrix multiplication divide and conquer algorithm.
The trick of the algorithm is reducing the number of multiplications to 7 instead of 8. I was wondering, can we ...
- 113
11
votes
1
answer
636
views
A simple proof for a theorem of Szekeres and Turán
Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
- 24.7k
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 111
11
votes
3
answers
1k
views
Congruence subgroups as abstract groups
This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup
$$
\pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:\;a,b,c,d\in\...
- 13k
10
votes
1
answer
441
views
Probability that a random distance function is metric
Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...
- 6,962
10
votes
1
answer
319
views
Construction of skew-Hadamard matrix of order 292
I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
- 261
10
votes
1
answer
2k
views
Arithmetic-geometric mean of positive matrices
Let $A,B$ be positive definite (Hermitian) matrices. Define the Arithmetic-geometric means of positive matrices by $A_0=A, G_0=B$, $A_{n+1}=\frac{A_n+G_n}{2}, G_{n+1}=A_n\natural G_n$, where $A_n\...
- 223
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
10
votes
2
answers
4k
views
Perturbation theory for the generalized eigenvalue problem
Is there a standard reference for the perturbation theory of the generalized eigenvalue problem?
More specifically, I would like to get a systematic expansion for the problem
$(A_0 + \epsilon A_1)...
- 1,193