**138**

votes

**19**answers

19k 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 ...

**63**

votes

**2**answers

4k 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 ...

**44**

votes

**7**answers

3k 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 + ...

**42**

votes

**1**answer

3k 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 ...

**40**

votes

**1**answer

1k 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 ...

**35**

votes

**1**answer

982 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 ...

**28**

votes

**2**answers

1k 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 ...

**27**

votes

**2**answers

2k 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 ...

**27**

votes

**7**answers

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 ...

**27**

votes

**3**answers

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 ...

**26**

votes

**1**answer

788 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\\\ ...

**25**

votes

**4**answers

2k 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 ...

**25**

votes

**1**answer

772 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 ...

**24**

votes

**2**answers

1k 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 ...

**21**

votes

**4**answers

24k 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 ...

**21**

votes

**4**answers

1k 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 ...

**21**

votes

**2**answers

672 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 ...

**21**

votes

**2**answers

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 ...

**20**

votes

**4**answers

1k 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 ...

**19**

votes

**10**answers

4k 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 ...

**19**

votes

**11**answers

3k 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 ...

**19**

votes

**4**answers

902 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

**1**answer

3k views

### How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. However, the constant is so large that ...

**18**

votes

**3**answers

1k views

### Is the semigroup of nonnegative integer matrices with determinant 1 finitely generated?

The group of $n\times n$ matrices with integer entries and determinant equal to 1, $SL(n,Z)$, is a finitely generated group (in fact, it is generated by 2 matrices). I am interested to know if the ...

**18**

votes

**2**answers

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 ...

**18**

votes

**4**answers

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 ...

**18**

votes

**3**answers

888 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, ...

**18**

votes

**1**answer

547 views

### Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector?

I'd like to state explicitly a problem which was somehow hidden in my three-week-old post:
Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal ...

**18**

votes

**0**answers

914 views

### 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 ...

**18**

votes

**0**answers

507 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 ...

**17**

votes

**6**answers

2k 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

**5**answers

886 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 ...

**17**

votes

**2**answers

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 $B$ ...

**17**

votes

**1**answer

860 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 ...

**17**

votes

**1**answer

1k views

### Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...

**17**

votes

**1**answer

336 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 $\mathbb{M}_m(\mathbb{M}_n)$ ...

**17**

votes

**1**answer

633 views

### Extremal points of the unit ball of M_n(R) …

The unit ball of ${\bf M}_n(\mathbb R)$ is a compact convex subset. As such, it is (Krein-Milman theorem) the convex envelop of its extremal points. So far, so good; but the unit ball depends of the ...

**16**

votes

**4**answers

2k 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 ...

**16**

votes

**3**answers

918 views

### Approximating commuting matrices by commuting diagonalizable matrices

Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that
$A_n \rightarrow A$, $B_n \rightarrow B$.
Each $A_n$ is diagonalizable and the same for ...

**16**

votes

**3**answers

2k views

### Alternate and symmetric matrices

Greetings to all !
Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO.
Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is ...

**16**

votes

**2**answers

410 views

### What is $A+A^T$ when $A$ is row-stochastic ?

This is motivated by this MO question.
If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is
symmetric,
entrywise ...

**16**

votes

**1**answer

1k views

### Commuting unitaries

Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...

**15**

votes

**4**answers

731 views

### Is there a matrix C so that the trace of C^n is dense in R?

I am looking for a matrix $C$ so that the sequence $tr(C^n)$ is dense in the set of real numbers. Equivalently (in the $2 \times 2$ case), find a complex number $z$ so that the sequence $z^n+w^n$ is ...

**15**

votes

**5**answers

783 views

### What fraction of n x n invertible integer matrices contain at least one unit?

The question is simple:
What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
I'm not sure what the correct measure ...

**15**

votes

**1**answer

813 views

### Analogue of Smith normal form for matrices over $\mathbb Z[t]$

Let $R$ be a principal ideal domain and $A \in M_n R$. It is well known that there exist invertible matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a_1,\dots,a_n)$ such that
$a_i \mid ...

**15**

votes

**2**answers

428 views

### ULU Decomposition of a matrix

Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...

**15**

votes

**1**answer

414 views

### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

**15**

votes

**3**answers

630 views

### Random products of projections: bounds on convergence rate?

The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...

**15**

votes

**1**answer

579 views

### Standard polynomials applied to matrices

The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots ...

**15**

votes

**0**answers

478 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 ...