All Questions
6,290 questions
8
votes
1
answer
1k
views
Determine if a matrix is unimodular
Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?
- 3,165
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
9
votes
4
answers
826
views
What is the minimal number of symmetric generators of the full matrix algebra?
Is there any lower bound known for the minimal number of generators needed to generate the full matrix algebra of real $n\times n$ matrices — when using only symmetric matrices for the generators?
...
5
votes
2
answers
495
views
Hadamard product and inertia
One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...
- 6,962
12
votes
3
answers
607
views
Example of a form linear in infinitely many variables ?
We all know plenty of examples of multilinear forms in finitely many variables (e.g. determinants). However, I am missing an interesting example of a form in infinitely many variables, linear in each. ...
- 4,492
10
votes
1
answer
2k
views
Over which fields is the Sylvester law of inertia valid?
Short version:
Over which fields is the (appropriate version of the) "Sylvester law of inertia" valid?
Long version:
Let $V$ be a finite dimensional vector space over the field $\Bbbk$ of ...
- 23.3k
2
votes
1
answer
566
views
Reference for a derivative formula for matrices
I found the identity
$$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$
On the matrix cookbook (http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf). It is equation ...
user7807
5
votes
2
answers
1k
views
Generalizations of Oppenheim's inequality
The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...
- 6,962
0
votes
1
answer
180
views
(probably simple) optimization question
Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
- 6,962
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
11
votes
1
answer
1k
views
Extending an assignment property from Q to R (or C)
Property of any odd number of nonnegative integers:
Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
- 7,831
1
vote
1
answer
212
views
name for a matrix operation
If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?
- 6,962
5
votes
2
answers
774
views
Can we calculate the inner product of a semicontinous function with the Dirac delta function?
Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
- 1,649
5
votes
0
answers
442
views
A reference on semisimple linear algebra
Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...
- 4,010
1
vote
1
answer
263
views
Is the trivial solution the only solution? [closed]
Let n be a positive integer, and c_1, c_2, ... c_n be (unkown) real numbers .
Consider the system
$$c_1+c_2+ ... +c_n=0,$$
$$c_1^2+c_2^2+ ... +c_n^2=0,$$
$$c_1^3+c_2^3+ ... +c_n^3=0,$$
$$.... .......
- 11
16
votes
1
answer
774
views
Minimizing the excursion of a sum of unit vectors
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...
- 151k
11
votes
5
answers
4k
views
Centralizer of a Matrix over a Finite Field
This question in stackExchange remained unanswered.
Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)...
- 801
0
votes
0
answers
324
views
Changing basis on an extension of a free Z-module.
Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in $. Then $c$ "incarnates"...
- 13.3k
12
votes
2
answers
759
views
When Do a Few Eigenvectors of Graph Laplacians Not Determine the Graph?
Essentially as the title, but I'll give a little bit more background.
I have some finite graph $G$ with $n$ vertices and adjacency matrix $A$. Let $D$ be the $n$ by $n$ matrix with the degree of ...
- 193
0
votes
0
answers
154
views
linsolve derivative
Consider a vector $\mathbf{g} \in \mathbb{R}^{m}$ and a matrix $\mathbf{A} \equiv \mathbf{A(g)} \in \mathcal{M}_{p\times q} \[\mathbb{R}\]$, a function of $\mathbf{g}$.
Furthermore, let $\mathbf{S} \...
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
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
9
votes
1
answer
894
views
Unusual decomposition of 3x3 real symmetric matrices - is this possible?
If $M$ is a 3x3, real symmetric matrix, then I know there are a few ways to decompose $M$ as
$M = A^T D A$,
where $D$ is a real diagonal matrix: e.g., this can always be done for some $A \in SO(3)$, ...
- 818
5
votes
1
answer
234
views
A conjecturally easier version of matrix Descartes rule of signs.
This is (quite obviously) inspired by this question. Let $C_i$ be symmetric positive definite matrices. Then is it true that there is exactly one symmetric positive definite $X$ such that
$F(X) = X^n ...
- 96.4k
10
votes
1
answer
4k
views
Eigenvalues, singular values, and the angles between eigenvectors
Suppose the $n \times n$ matrix $A$ has eigenvalues $\lambda_1, \ldots, \lambda_n$ and singular values $\sigma_1, \ldots, \sigma_n$. It seems plausible that by comparing the singular values and ...
- 571
2
votes
2
answers
561
views
The number of solutions of a matrix equation
Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ be a polynomial, $a_i \in \mathbb{R}$ for all $i$. Set
$$S = \lbrace A \in \mathbb{M}_n: P(A) = 0 \rbrace.$$
We consider the following relation $\sim$ on $S$:...
- 2,773
0
votes
0
answers
204
views
Matrix Mutiplication through Matrix Logarithms and Exponentials
Let $A,B$ be full rank $n \times n$ matrices. If $AB = BA$, then $\exp(\log(A)+\log(B))=AB$.
Supposing $A = USL$ and $B = VSL$ where $U,V,S,L$ are integer valued matrices, $det(L)=1$ and $U = LVL^{-1}...
- 13.9k
7
votes
0
answers
355
views
An $L^{\infty}$ version of principal component analysis?
I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
- 193
1
vote
0
answers
109
views
can minimal volume rational subspaces in a lattice be arbitrarily 'close'.
Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: ...
- 2,274
5
votes
1
answer
713
views
Invertible matrix
Let $K$ be a field s.t. $charac(K)\not= 2$. Let $A\in\mathcal{M}_n(K)$ be such that $rank(A)\geq n/2$. Can one find a matrix $B$ such that $B$ is similar to $A$ and $A+B$ is invertible ?
- 3,741
6
votes
1
answer
771
views
Eigenvalues of reverse bidiagonal matrices
Is there an easy way of determining if the eigenvalues of a real-valued reverse bidiagonal matrix are real. Basically I have two vectors $(a_1,...,a_n)$ and $(b_1,...,b_{n-1})$ that form the "reverse" ...
- 61
3
votes
0
answers
395
views
Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$
Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That ...
- 5,327
2
votes
3
answers
1k
views
On certain decomposition of unitary symmetric matrices
This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here.
It is well known that a symmetric matrix over ...
- 355
5
votes
1
answer
320
views
Two Concepts of Monotonicity
Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that
$F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that
$$
\langle F(y)-F(x), y-x\...
- 51
9
votes
1
answer
611
views
Matrix-tree theorem via supersymmetry (i.e. Grassman algebras)
The matrix-tree theorem states the number of spanning trees of a graph $G$ is equal to a modified determinant of the adjacency matrix or "graph Laplacian", $\Delta_G$:
$$\#\{ \text{spanning ...
- 22.8k
1
vote
1
answer
1k
views
trace of a matrix of finite order
Let $A$ be an $n$ by $n$ real matrix of order $d$. i.e. $d$ is the smallest positive integer greater than $1$ that makes $A^{d}=I_{n}$.
The set of trace zero real matrices form $n^{2}-1$ dimensional ...
- 133
2
votes
1
answer
187
views
Partial isometries making families of linearly independent vectors orthogonal
Suppose I have a family of $n$ linearly-independent elements $v_i$ of the Hilbert space $\mathbb{C}^m$, which are not necessarily orthogonal. Can I always find a partial isometry $f: \mathbb{C} ^m \to ...
- 2,513
8
votes
3
answers
431
views
Characterising semi-definite positiveness on vectors with non-negative entries
My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
- 1,352
2
votes
1
answer
118
views
Dual of a semilinear morphism
Let $R$ be a commutative ring and let $M$ and $N$ be $R$-modules. Let $\sigma:R\rightarrow R$ be a ring automorphism.
Let $f: M\rightarrow N$ be a $\sigma$-semilinear map, i.e. a map of abelian ...
- 11.1k
15
votes
1
answer
777
views
Reconstructing a word
Let $w(a,b)$ be a word in two letter alphabet. Let $$A=\left(\begin{array}{lll}x_1 & x_2 & x_3\\\ x_4 &x_5 & x_6\\\ x_7 & x_8 & x_9\end{array}\right), B=\left(\begin{array}{lll}...
user6976
7
votes
2
answers
2k
views
Urge/reason for inventing interior product ( Grassmann algebra )
Hello everyone,
I wanted to lecture on Grassmann and his works , and I have been reading the collected works of Grassmann " Die Lineale Ausdehnungslehre ". There Grassmann introduced something ...
3
votes
2
answers
2k
views
How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?
Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V_s$...
- 4,907
5
votes
2
answers
2k
views
Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is
$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$.
In ...
- 846
3
votes
1
answer
533
views
Solving a system of linear inequalities
Consider the following system of inequalities:
$Ax=b$;
$x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without ...
- 221
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
0
votes
1
answer
193
views
Ease of calculation of norm
I have SPD matrix A and two vectors z and b.
Is there exist a norm where I can calculate $||A^{1/2}b-z||$ without having to calculate $A^{1/2}b$ explicitly ?
- 489
4
votes
2
answers
2k
views
Generalized Eigenvalue Problem with one matrix having low rank
I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP.
The Problem
Let $A$ be ...
- 41
1
vote
2
answers
393
views
Have derivatives of determinants along 1-psg's ever been 'coherently' computed via Jacobi's formula?
Suppose $\mathfrak{p}$ denotes all the symmetric matrices in $\mathfrak{sl}_{2n} \mathbb{R}$.
Then for each parameterized 1-dimensional linear subspace $\xi=\xi(t)$ of $\mathfrak{p}$ we get a 1-...
- 2,274
11
votes
4
answers
4k
views
When is the infimum of an arbitrary family of measurable functions also measurable?
Let $(X,\Sigma,\mu)$ be a measure space and consider a family of $\mu$-measurable functions $f_i:X \to \mathbb{R}$ for $i$ lying in some index set $I$. Define $$f(x) = \inf_{i \in I} f_i(x)$$
I think ...
- 15.5k
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