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Questions tagged [matrices]

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 tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

237
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25answers
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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 should ...
87
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9answers
16k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\frac{1}{ad - bc} \begin{pmatrix} d & -b \\\ -c & a \end{...
71
votes
3answers
6k 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 $\...
56
votes
7answers
6k views

How to prove this determinant is positive?

Given 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 the following? $$\det \big( I + e^...
55
votes
1answer
5k views

Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
51
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4answers
52k 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 the matrices are Hermitian and positive definite? I am ...
48
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1answer
2k views

Invertible matrices over noncommutative rings

Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples? The question popped up ...
44
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1answer
3k views

Determinant of a determinant

Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m \...
38
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3answers
3k 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=...
36
votes
6answers
36k views

Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that $$\det(A+B) \ge \det(A) + \det(B)$$ in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
35
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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 ...
33
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2answers
11k views

How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). ...
30
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7answers
2k 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 ...
29
votes
3answers
2k views

Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
28
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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 ...
27
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12answers
5k views

When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...
27
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1answer
2k views

Is there an explicit formula for the hessian of “Determinant”?

Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function. Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
27
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1answer
993 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of length $...
27
votes
2answers
2k views

Invertible matrices satisfying $[x,y,y]=x$

I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope. Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and $[x,y,...
27
votes
1answer
888 views

Understanding “infinite” relations in groups

Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ 1&2&2\end{...
26
votes
10answers
6k views

When to pick a basis?

Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the ...
26
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4answers
2k views

Matrices: characterizing pairs $(AB, BA)$

Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
26
votes
0answers
346 views

Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?

For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...
25
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1answer
485 views

Determinants of binary matrices

I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
25
votes
2answers
528 views

Symmetric strengthening of the Cauchy-Schwarz inequality

In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle ...
24
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5answers
1k views

When is a matrix power nonnegative

The following question came up today during a discussion: Suppose $A$ is an $n \times n$ real matrix. Is there some way to tell whether there exists an integer $q > 0$ such that $A^q$ is ...
24
votes
2answers
866 views

Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation: http://www.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
24
votes
2answers
851 views

Factorization of a real matrix into Hermitian x Hermitian. Is it stable ?

It is known (see Theorem 4.1.7 in R. Horn & C. Johnson) that every matrix $A\in M_n(\mathbb R)$ (real entries) can be written as the product $HK$ of two Hermitian matrices (complex entries). Of ...
24
votes
0answers
745 views

Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
23
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3answers
1k views

On determinants formed by binomial coefficients

Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers. Let us form naively a $q \times q$ ...
23
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3answers
1k views

Is every positive integer the permanent of some 0-1 matrix?

In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely: Is it true that for every positive integer $k$ there exists a balanced ...
23
votes
2answers
2k views

Finite subgroups of unitary groups

Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite ...
22
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6answers
3k views

Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n x^n$....
22
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6answers
1k views

Cayley-Hamilton revisited

Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let $...
22
votes
5answers
8k views

Deriving inverse of Hilbert matrix

The Hilbert matrix is the square matrix given by $$H_{ij}=\frac{1}{i+j-1}$$ Wikipedia states that its inverse is given by $$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...
22
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2answers
8k views

Infinite matrices and the concept of “determinant”

Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
21
votes
1answer
2k 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 ...
21
votes
6answers
2k views

What are the possible eigenvalues of these matrices?

Edit: since we seem a bit deadlocked at this point, let me weaken the question. It's fairly easy to see that the set of 8-tuples of reals which can be the eigenvalues of a matrix of the desired form ...
21
votes
4answers
1k views

A matrix algebra has no deformations?

I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about finite-...
21
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2answers
2k views

Singular values of sequence of growing matrices

I asked this question on math.stackexchange and haven't received an answer in two weeks, so I'm repeating it here. Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \cr 1/2 & 0 &...
21
votes
4answers
3k views

Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
21
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0answers
695 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
21
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0answers
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The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
20
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6answers
5k views

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

I asked this (with background) here https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision but did not really get any answers. ...
20
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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 \...
20
votes
2answers
2k views

Integer matrices with no integer eigenvalues

Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
20
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2answers
1k views

Lifting matrices mod 2 to integers.

The following question was motivated by my research. Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
20
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3answers
1k views

Simultaneous “orthonormalization” in $\mathbb{C}^4$

Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix. So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
20
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0answers
4k views

An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?

A famous result in linear algebra is the following. An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$. I know one proof using the Smith Normal Form (SNF). ...
20
votes
0answers
760 views

Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by \begin{equation*} e_k(\vx) := \sum_{1 \...