All Questions
629 questions
38
votes
1
answer
10k
views
Infinite tensor products
Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
- 63.1k
91
votes
5
answers
124k
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 ...
- 1,013
368
votes
31
answers
80k
views
Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...
57
votes
6
answers
6k
views
Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?
If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
- 3,511
20
votes
6
answers
42k
views
Eigenvalues of symmetric tridiagonal matrices
Suppose I have the symmetric tridiagonal matrix:
$$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & \ddots & \vdots \\\
0 & b_{2} & a & \...
47
votes
4
answers
8k
views
Does the fact that this vector space is not isomorphic to its double-dual require choice?
Let $V$ denote the vector space of sequences of real numbers that are eventually 0, and let $W$ denote the vector space of sequences of real numbers. Given $w \in W$ and $v \in V$, we can take their "...
- 3,982
42
votes
3
answers
5k
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=...
- 52.3k
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
127
votes
4
answers
32k
views
Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional
A very important theorem in linear algebra that is rarely taught is:
A vector space has the same dimension as its dual if and only if it is finite dimensional.
I have seen a total of one proof of ...
68
votes
4
answers
9k
views
explicit big linearly independent sets
In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or ...
- 63.1k
29
votes
6
answers
10k
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. ...
- 2,600
15
votes
2
answers
835
views
What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
- 26.5k
10
votes
4
answers
2k
views
Formal power series is Taylor expansion of rational function iff Hankel determinants vanish?
Let $$ u(T)=\sum_{n = 0}^\infty a_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $...
user97565
4
votes
5
answers
4k
views
About adding a negative definite rank-1 matrix to a symmetric matrix
If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...
- 1,893
81
votes
10
answers
9k
views
Existence of a zero-sum subset
Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:
Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...
- 85.6k
53
votes
7
answers
51k
views
Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\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$-...
- 541
45
votes
11
answers
23k
views
real symmetric matrix has real eigenvalues - elementary proof
Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
- 579
34
votes
2
answers
2k
views
What are the algebras for the double dualization monad?
Let $k$ be a field, and let $\mathbf{Vect}$ denote the category of vector spaces (possibly infinite-dimensional) over $k$. Taking duals gives a functor $(\ )^*\colon \mathbf{Vect}^{\mathrm{op}} \to \...
- 27.7k
28
votes
5
answers
4k
views
Does Smith normal form imply PID?
Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...
user5810
28
votes
4
answers
5k
views
Jacobi's equality between complementary minors of inverse matrices
What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...
- 4,466
26
votes
2
answers
3k
views
Sizes of bases of vector spaces without the axiom of choice
Assuming the axiom of choice does not hold we have that there is a vector space without a basis. The situation can be, in some sense, worse. It is consistent that there are vector spaces that have two ...
- 39.7k
25
votes
1
answer
4k
views
What kind of random matrices have rapidly decaying singular values?
I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
- 118k
25
votes
4
answers
7k
views
"Natural" pairings between exterior powers of a vector space and its dual
Let $V$ be a finite-dimensional vector space over a field $k$, $v_1, \dotsc v_n \in V$ a set of vectors, and $f_1, \dotsc f_n \in V^{\ast}$ a set of covectors. Up to permutation, there seem to be at ...
- 118k
20
votes
2
answers
1k
views
Euler numbers and permanent of matrices
Motivated by Question 402249 of Zhi-Wei Sun, I consider the permanent of matrices
$$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]_{1\le j,k\le n-1},$$
where $n$ is ...
- 884
17
votes
2
answers
4k
views
Optimizing the condition number
Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...
- 96.4k
16
votes
2
answers
9k
views
Solving a quadratic matrix equation
This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? ...
- 1,538
9
votes
4
answers
2k
views
Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?
I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.
Background
A simple consequence of the singular value decomposition is that any vector $...
- 846
4
votes
2
answers
1k
views
Reference request: Oldest linear algebra books with exercises?
Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
3
votes
1
answer
252
views
Is $\rho(X_1\dots X_r)^{2/r}\leq \frac{d}{r}\cdot\rho(X_1\otimes X_1+\dots+X_r\otimes X_r)$ for $d\times d$-real matries $X_1,\dots,X_r$?
Let $\rho(A)$ denote the spectral radius of a square matrix $A$. Let $r,d$ be positive integers. Let $X_1,\dots,X_r$ be $d\times d$-real matrices. Then do we necessarily have $$\rho(X_1\dots X_r)^{2/r}...
2
votes
1
answer
184
views
Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?
Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
- 24.9k
128
votes
13
answers
27k
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 $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
- 7,337
54
votes
5
answers
2k
views
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
- 10.1k
53
votes
9
answers
13k
views
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,306
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
43
votes
6
answers
9k
views
"A gentleman never chooses a basis."
Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
Question. Is there a gentlemanly way to prove that the natural map from $V$ to $V^{**}$ is surjective if $V$...
- 5,275
33
votes
2
answers
7k
views
Dimension of infinite product of vector spaces
This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...
- 20.8k
25
votes
3
answers
4k
views
Largest number of vectors with pairwise negative dot product
What is the largest $m$ such that there exist $v_1,\dots,v_m \in \mathbb{R}^n$ such that for all $i$ and $j$, $1\leq i< j\leq m$, we have $v_i \cdot v_j < 0$.
Also, the preview screen is not ...
user5810
21
votes
3
answers
1k
views
Which doubly stochastic matrices can be written as products of pairwise averaging matrices?
A matrix $A$ is called doubly stochastic if its entries are nonnegative, and if all of its rows and columns add up to $1$. A subset of doubly stochastic matrices is the set of pairwise averaging ...
- 415
19
votes
3
answers
6k
views
What are the matrices preserving the $\ell^1$-norm?
So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
- 391
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
12
votes
3
answers
784
views
Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 128k
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
11
votes
1
answer
579
views
Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
- 43.2k
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
10
votes
5
answers
3k
views
Generalization of the polarisation formula for symmetric bilinear forms to symmetric multilinear forms
Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vector space and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$.
The Polarisation Formula ...
- 109
10
votes
1
answer
5k
views
Eigenvalues of the sum of a diagonal and a unit matrix
I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that
$A = D + J$
Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
...
- 293
9
votes
0
answers
359
views
Factorisation of a polynomial from the Boolean algebra
Let $B_n$ denote the Boolean algebra of a set with $n \geq 2$ elements and $C_n$ the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else, where $i,j\in B_n$.
Let $M_n:=C_n+C_n^T$ and $...
- 26.5k
8
votes
1
answer
1k
views
Why does an invertible complex symmetric matrix always have a complex symmetric square root?
Chapter XI Theorem 3 from here implicitly states that an invertible complex symmetric matrix always has a complex symmetric square root.
It's clear that a square root exists, by appealing to the ...
- 4,943
7
votes
1
answer
969
views
The height of the Perron-Frobenius eigenvector
Does the height of a real symmetric matrix with non-negative entries control the height of its Perron-Frobenius eigenvector, under some reasonable definition of heights?
Just as an example of what ...
- 23k
7
votes
2
answers
4k
views
Number of Plücker relations for a Grassmannian
Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: $$p_{...
- 349