All Questions
Tagged with linear-algebra reference-request
318 questions
47
votes
3
answers
3k
views
Why is the Vandermonde determinant harmonic?
It can be checked that the Vandermonde determinant defined as
$$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$
is a harmonic function, that is $\Delta V = 0$ where ...
- 763
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
38
votes
10
answers
18k
views
Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
36
votes
2
answers
32k
views
Eigenvalues of the product of two symmetric matrices
This is mostly a reference request, as this must be well-known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
- 2,600
35
votes
4
answers
2k
views
Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?
A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...
- 2,527
34
votes
4
answers
2k
views
If $A,B$ are upper triangular matrices such that $AX=XA\implies BX=XB$ for upper triangular $X$, is $B$ a polynomial in $A$?
A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here.
...
- 343
31
votes
1
answer
2k
views
solving linear equations made difficult
(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.)
I saw this amusing derivation ...
- 19.7k
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
27
votes
6
answers
6k
views
Origin of exact sequences
I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
- 443
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 387
27
votes
3
answers
13k
views
What is known about the distribution of eigenvectors of random matrices?
Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...
- 433
26
votes
1
answer
1k
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 ...
- 5,186
23
votes
2
answers
3k
views
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
- 493
20
votes
2
answers
1k
views
Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative
I saw this problem some years ago and I would greatly appreciate any reference or solution.
Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
- 3,153
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
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
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
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
3
answers
3k
views
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
4
answers
10k
views
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
16
votes
4
answers
1k
views
Reference for a linear algebra result
I asked the following question (https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no ...
- 26.9k
16
votes
4
answers
3k
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 ...
- 1,795
16
votes
3
answers
791
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 ...
- 181
15
votes
4
answers
869
views
What is known about ordinary character values at involutions?
Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...
- 1,755
15
votes
4
answers
4k
views
Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?
When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...
- 403
14
votes
4
answers
3k
views
Vandermonde matrix is totally positive
A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
- 5,428
14
votes
1
answer
417
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
13
votes
2
answers
915
views
Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 674
13
votes
3
answers
2k
views
Linear algebra underlying quantum entanglement?
Hope this question is appropriate. I think I saw certain claims that quantum entanglement is a certain phenomena that can be explained (or modelled) in terms of tensor products in linear algebra. I ...
- 14.3k
13
votes
4
answers
2k
views
Groups of matrices in which all elements have all eigenvalues equal in modulus
I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
- 6,206
13
votes
1
answer
1k
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 ...
- 6,206
13
votes
0
answers
237
views
A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
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
2k
views
Is there an eigenvalue of modulus larger than 1?
Given a matrix $A\in \operatorname{SL}_d(\mathbb{Z})$ (the special linear group) satisfying the two conditions: (1) no eigenvalue of $A$ is a root of unity, (2) the characteristic polynomial of $A$ is ...
- 43.2k
12
votes
3
answers
2k
views
Representability of matroids over $\mathbb R$
Let $M$ be a matroid, for example viewed as being given by a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$, $d(\lbrace x \rbrace)=1$, for all $x \in X$,...
- 25.5k
12
votes
2
answers
1k
views
Quadratic Farkas' Lemma?
The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...
- 23k
12
votes
2
answers
4k
views
How can one construct a sparse null space basis using recursive LU decomposition?
Given an $m$ by $n$ matrix $A$ I'm familiar with the standard method to compute a basis for the null space of $A$ by computing a QR factorization of $A^T$. If $A$ is large and sparse, we can use ...
- 723
11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
- 43.2k
11
votes
3
answers
1k
views
Diagonalization via the Toda flow
According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalised via the Toda flow. More specifically, if $X=X(t)\in\...
- 2,933
11
votes
1
answer
1k
views
A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 5,380
11
votes
2
answers
820
views
Encyclopedia of properties of nonnegative matrices
I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $a_{ij} \ge 0$ for all $1 \le i,j \le n$.
Chapter 8 ...
- 682
11
votes
1
answer
633
views
How do computer algebra packages like Sagemath implement rank of a matrix
I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using ...
- 263
11
votes
1
answer
731
views
Reference request: Volume 2 of Abhyankar's lectures on algebra?
Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra.
The description:
This book is a timely survey of much of the algebra developed ...
- 111
10
votes
3
answers
830
views
Find the inverse of a matrix that is very similar to the Hilbert matrix
The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely,
$$...
- 153
10
votes
4
answers
8k
views
Any reference on multilinear algebra [closed]
Do you know any good reference on multilinear algebra?
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
10
votes
1
answer
262
views
What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$?
Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are ...
- 7,633
10
votes
1
answer
520
views
Homogeneous polynomials, mixed determinants, positive definiteness
Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial
$$
f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n})
$$
never vanishes on $\...
- 3,946