Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level ...

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28
votes
4answers
32k views

Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when they are Hermitian and positive-definite? I am investigating ...
14
votes
3answers
3k views

Number of unique determinants for an NxN (0,1)-matrix.

I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore won't have a determinant. While it might also be ...
171
votes
22answers
25k views

Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ...
30
votes
2answers
2k views

The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
26
votes
4answers
3k views

Adjacency matrices of graphs

Motivated by the apparent lack of possible classification of integer matrices up to conjugation (see here) and by a question about possible complete graph invariants (see here), let me ask the ...
11
votes
5answers
1k views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
15
votes
7answers
6k views

Expected determinant of a random NxN matrix.

What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?
10
votes
2answers
3k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
14
votes
4answers
985 views

Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 &...
12
votes
5answers
2k views

Bounding the absolute sum of entries of the inverse of a 0-1 matrix

I have a non-singular square 0-1 matrix and I want to bound the sum of absolute values of its inverse as a function of n (or the vector 1-norm). Asymptotic results are also useful. Does anyone know ...
7
votes
2answers
583 views

Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set $$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$ Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by $$d(A,B)=\sqrt{1-\...
5
votes
0answers
257 views

concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
9
votes
4answers
2k views

Is any $(n-1)\times (n-1)$ submatrix of an $n \times n$ Vandermonde matrix invertible?

Given an $n \times n$ vandermonde matrix $V$ which is invertible, is any $(n-1) \times (n-1)$ submatrix of $V$ invertible also? I think the answer is yes, but I don't know how to prove.
12
votes
1answer
491 views

A maximal element, where Schur gives a minimal element

Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question Hadamard-like inequalites for positive definite symmetric matrices. If $H$ is a subgroup of $\frak ...
1
vote
5answers
336 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
10
votes
2answers
2k views

Eigenvalues of nonnegative integer matrices

Edit I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post: What are the possible eigenvalues of nonnegative integer matrices? Any answer ...
7
votes
1answer
758 views

The saturation of Murray von Neumann relation

Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version: Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows: We say $a M b$ iff $...
3
votes
2answers
198 views

functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...
3
votes
1answer
205 views

Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
1
vote
0answers
377 views

Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected. The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
1
vote
1answer
210 views

A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function. What matrices belongs to $S$, precisely? Let $M=...
64
votes
2answers
5k views

Norms of Commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
49
votes
7answers
4k views

How to prove this determinant is positive?

Given the matrices $ A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr) $, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + e^{A_1}e^{A_2}\...
31
votes
3answers
3k views

A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here). Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
18
votes
4answers
3k views

Condition for two matrices to share at least one eigenvector?

Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so $...
13
votes
3answers
1k views

Distribution of the spectrum of large non-negative matrices

This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials. When considering (entrywise) ...
11
votes
3answers
2k 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 ...
12
votes
3answers
522 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 ...
27
votes
7answers
1k views

Does the truth of any statement of real matrix algebra stabilize in sufficiently high dimensions?

This question is related to this recent but currently unanswered MO question of Ricky Demer, where it arose as a comment. Consider the structure $R^n$ consisting of $n\times n$ matrices over the ...
18
votes
3answers
939 views

looking for proof or partial proof of determinant conjecture

Math people: I am looking for a proof of a conjecture I made. I need to give two definitions. For distinct real numbers $x_1, x_2, \ldots, x_k$, define $\sigma(x_1, x_2, \ldots, x_k) =1$ if $(x_1, ...
10
votes
3answers
872 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\...
8
votes
4answers
3k views

Eigenvectors and eigenvalues of Tridiagonal matrix

Hi, is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal matrix of the form : $$ \mathcal{T}^{a}_n(p,q) = \begin{pmatrix} 0 & q & 0 & 0 &\...
15
votes
4answers
1k views

Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?

I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{Tr(A^* A)/n}$. My question is whether a $k$-uple of hermitian matrices that are almost commuting (with ...
9
votes
3answers
1k 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 ...
9
votes
1answer
1k 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\...
8
votes
2answers
6k views

Fast trace of inverse of a square matrix

Which would be the most efficient way (in computational time) to compute tr(inv(H)), where H is a (dense) square matrix? In my particular problem I also have a LU decomposition of H already available,...
7
votes
3answers
2k views

2-norm of the upper triangular “all-ones” matrix

Let $M_n$ be the $n\times n$ matrix $$ (M_n)_{ij}=\begin{cases}1 & i\leq j,\\\\ 0 &i>j.\end{cases} $$ Is there around an explicit expression or at least an asymptotic for $\left\Vert M_n \...
20
votes
4answers
1k views

Does Anyone Know Anything about the Determinant and/or Inverse of this Matrix?

The matrix I am inquiring about here is the $n \times n$ matrix where the entry $A_{ij}$ is $\frac{1}{(i+j-1)^2}$. The $2 \times 2$ matrix looks like $$ \begin{pmatrix} 1 & 1/4 \\ 1/4 & 1/9 \...
18
votes
6answers
3k views

how to find/define eigenvectors as a continuous function of matrix?

I asked this (with background) here http://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision but did not really get any answers. ...
17
votes
1answer
402 views

A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$ I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
9
votes
3answers
493 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
9
votes
4answers
1k views

How many minors I need to check to conclude all minors will vanish ?

Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish. I remember hearing that one really does not need to check all possible minors in order to conclude ...
8
votes
0answers
458 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 ...
3
votes
1answer
792 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that $$\sum_{i=1}^...
14
votes
1answer
1k views

maximal order of elements in GL(n,p)

I am looking for a formula for the maximal order of an element in the group $\operatorname{GL}\left(n,p\right)$, where $ p$ is prime. I recall seeing such a formula in a paper from the mid- or early ...
13
votes
1answer
600 views

Free subgroups of GL(2,Z)

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle < {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, ...
13
votes
1answer
1k views

How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

I asked this question in Math Stack Exchange earlier here: http://math.stackexchange.com/questions/1199380/what-is-the-intuition-behind-how-can-we-interpret-the-eigenvalues-and-eigenvec and since I ...
12
votes
0answers
566 views

Is “being a full ring of quotients” a Morita invariant property?

Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
11
votes
1answer
472 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 ...
9
votes
4answers
857 views

Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation. Since I essentially need $n\le 4$, I think that I can show it ...