All Questions
6,026 questions
19
votes
2
answers
1k
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Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
- 18.4k
19
votes
1
answer
494
views
Linear maps between arbitrarily chosen vectors of vector spaces $V$ and $W$
I recently came across this question:
Is the axiom of choice needed to prove the following statement:
Let $V, W$ be vector spaces, and suppose $V \neq \{0\}$. Let $v \in V$, $v \neq 0$, $w \in W$. ...
- 191
19
votes
1
answer
692
views
What is the groupoid cardinality of the category of vector spaces over a finite field?
For any groupoid, it's groupoid cardinality is the sum of the reciprocals of the automorphism groups over the isomorphism classes. Let us consider the category of vector spaces over a finite field $\...
- 7,746
19
votes
3
answers
2k
views
Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
19
votes
4
answers
2k
views
Variation on a matrix game
The original problem appeared on last year's Putnam exam:
"Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays first. At each turn, ...
- 1,975
19
votes
1
answer
4k
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How should I think about the module of coinvariants of a $G$-module?
Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...
- 7,527
19
votes
1
answer
856
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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 $\mathbb{M}_m(\mathbb{M}_n)$ ...
- 1,748
19
votes
1
answer
948
views
Recognize this strange expression from linear algebra?
I've come across an odd-looking expression and oh how I wish I had a more elegant description of it. Maybe someone who enjoys symmetric bilinear forms in characteristic two will recognize it? Or ...
- 13.3k
19
votes
1
answer
904
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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 M}_{m\...
- 23k
19
votes
1
answer
895
views
Are the only local minima of $\angle(v, Av)$ the eigenvectors?
Let $A$ be an invertible $n \times n$ complex matrix. For $v \in \mathbb{CP}^{n-1}$, define
$$d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle \...
- 156k
19
votes
1
answer
825
views
Number of matrices with given Smith normal form
Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
- 1,101
18
votes
3
answers
8k
views
Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
- 32.1k
18
votes
2
answers
1k
views
Is a matrix similar to its transpose over $\mathbb{Z}_p$?
Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...
- 2,242
18
votes
1
answer
1k
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A linear algebra problem in positive characteristic
Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
- 4,484
18
votes
6
answers
6k
views
Computing signature
I have a feeling that this might have already been asked, but can't find the question. Anyway, the question is: given a symmetric $n\times n$ matrix, is there a faster way to compute its signature ...
- 96.4k
18
votes
3
answers
2k
views
Elementary $\mathrm{Ext}^1$ intuition
$\DeclareMathOperator{\Hom}{\operatorname{Hom}}\DeclareMathOperator{\Ext}{\operatorname{Ext}}$I am wondering what sort of basic basic intuitive meaning $\Ext^1(M,N)$ has.
As a base case: if $M$ and $N$...
- 411
18
votes
1
answer
635
views
Is Carlitz's paper correct about the number of similarity classes of commuting matrices?
L. Carlitz has a paper, Classes of pairs of commuting matrices over a finite field, that computes the number of simultaneous similarity classes of of pairs of commuting matrices in $\operatorname{Mat}...
- 472
18
votes
1
answer
847
views
Showing that a certain matrix is not positive definite
Let $J_k$ be a $k \times k$ all ones matrix and $B$ any $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$.
I would like to show that the matrix $$X_B = (J_k -I) - B (J_k - I)^{-...
- 3,463
18
votes
3
answers
2k
views
Torsion in GL_n(Z)
Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map ...
- 44.8k
18
votes
3
answers
6k
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 will trivially have zero determinant. While it ...
- 359
18
votes
2
answers
5k
views
Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
- 345
18
votes
3
answers
1k
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Example of a space for which $V \cong Hom(V,V)$
Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology.
Is there a non-trivial ...
- 8,512
18
votes
2
answers
3k
views
Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
- 54.6k
18
votes
2
answers
488
views
Encoding primes via ranks of sign matrices
(Reposted from math.SE)
Recently I came across a very simply defined family of matrices: for $n \in \mathbb{N}$, set $A_n := (a_{ij})_{0 \le i, j \le n-1}$, where
$$\displaystyle a_{ij} := (-1)^{\big\...
- 281
18
votes
1
answer
1k
views
How fast can extreme eigenvalues of the average of random matrices converge to their expectation?
Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
- 181
18
votes
1
answer
1k
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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 ...
- 4,705
18
votes
2
answers
1k
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Karoubi versus Kasparov K-theory
I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$
with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ for $i=1,\ldots,p$, ...
- 20.9k
18
votes
1
answer
1k
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A curious eigenvalue inequality
Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...
- 1,748
18
votes
0
answers
571
views
Fundamental Theorem of Algebra via multiple integrals
Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
- 60.5k
18
votes
0
answers
540
views
A curious switch in infinite dimensions
Let $V$ be a finite dimensional real vector space. Let $GL(V)$ be the set of invertible linear transformations, and $\Phi(V)$ be the set of all linear transformations. We can also characterize $\Phi(V)...
- 7,583
17
votes
2
answers
2k
views
The Lefschetz operator
Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2(\mathbb{R}^{2n})^*$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...
- 28k
17
votes
6
answers
3k
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Does the linear automorphism group determine the vector space?
I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...
- 12.9k
17
votes
2
answers
911
views
Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?
Is it possible to find the determinant of an $n\times n$- matrix, only given the determinant of all $p\times p$ sub-matrices in it? Here $p\leq n$ is fixed. This is obviously true if $p=1,n$. But what ...
- 293
17
votes
2
answers
750
views
Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?
Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism?
More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d&...
- 173
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votes
2
answers
2k
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Is the determinant the only multiplicative matrix function? [closed]
Is there a matrix invariant or property that is multiplicative, i.e.,
$$f(AB) = f(A) f(B)$$
other than the determinant? In addition, some matrix norms are submultiplicative, but is there a ...
- 1,465
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
4k
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Geometric interpretations of matrix inverses
$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (...
- 693
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3
answers
2k
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Finding the nearest matrix with real eigenvalues
In this thread on MATLAB Central, I found a discussion on finding the nearest matrix with real eigenvalues. The first hypothesis was to simply truncate the complex part of the eigenvalues. So, given ...
- 185
17
votes
4
answers
2k
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interlacing roots/eigenvalues results and modern analogues
Is there any relation between these theorems on interlacing roots?
The roots of $f(x), f'(x)$ interlace (if all the roots of $f(x)$ are real and have real coefficients).
The eigenvalues of an $n \...
- 22.8k
17
votes
3
answers
3k
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Linear algebra from the categorical point of view
Is there any book of Linear algebra in the modern language of Category theory?
I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...
- 545
17
votes
1
answer
668
views
A coincidence or a fact: determinants of two matrices
While playing around with the MO question Determinant with factorials is not 0? about a determinant of the Hankel matrix of entries $(i+j-2)!$, having the value $\prod_{k=0}^{n-1}k!^2$, I stumbled on ...
- 43.2k
17
votes
1
answer
1k
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How many values determine a norm?
It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely.
How many values do we need to specify in order to ...
- 6,741
17
votes
1
answer
3k
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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
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
17
votes
2
answers
4k
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Optimizing the condition number
Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...
- 96.4k
17
votes
1
answer
963
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Examples of vector spaces with bases of different cardinalities
In this question Sizes of bases of vector spaces without the axiom of choice it is said that "It is consistent [with ZF] that there are vector spaces that have two bases with completely different ...
- 173
17
votes
1
answer
2k
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Subgroups of $\mathbb{Z}^n$
I hope that the following problem isn't actually elementary (at least, for the sake of the fact that I'm posting it here), and I apologize if it is. I did try hard to solve it first.
Let $V$ be a $\...
- 1,786
17
votes
2
answers
1k
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Constructive proof of a rational version of Perron-Frobenius?
In the following, we work with vectors and matrices whose entries are rational numbers. Inequalities between such vectors are understood to be coordinatewise: e.g., two vectors $a = \left(a_1,a_2,\...
- 33.8k
17
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1
answer
4k
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How complicated is infinite-dimensional "undergraduate linear algebra"?
The name "undergraduate linear algebra" in the title is a bit of a joke, and so I don't know how widely spread it is. To wit:
High school linear algebra is the theory of a finite-dimensional vector ...
- 54.6k
17
votes
4
answers
10k
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Prime/undecomposable matrices
Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
- 2,855