All Questions
6,289 questions
4
votes
2
answers
1k
views
Vandermonde matrices and general position
I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of
columns, i.e. ...
- 41
6
votes
3
answers
7k
views
Calculating the Perron-Frobenius eigenvector of a positive matrix from limited information
In the background of this question is a matrix $A$, all of whose elements are positive. The Perron-Frobenius theorem tells us that the eigenvalue with largest absolute value is real, and that there ...
- 430
1
vote
0
answers
285
views
Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?
Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and $\mathbf{p}=[p_1\;\ldots\;p_m]^\top,\mathbf{q}=[q_1\;\ldots\;...
- 607
2
votes
1
answer
162
views
'Compute' Integral equivalence of matrices
Hi.
For a matrix $D \in \mathbb{Z}^{n \times n}$ and a symmetric, positive definite integral even matrix $S \in \mathbb{Z}^{n \times n}$ put $S[D] := D^TSD$ where the $\cdot^T$ means 'transposed'. ...
- 303
7
votes
3
answers
780
views
the largest eigenvalue of the matrix A with A_{ij}=(i \times j) mod p for p is a prime.
For a prime p, consider the $(p-1) \times (p-1)$ matrix A with entry to be $A_{ij}=(i \times j) mod$ $p$. every row (column) is permutation of 1 to p-1, such a permutation is useful in one version of ...
- 637
4
votes
1
answer
682
views
Canonical rational form for $SL(n)$
The canonical rational form helps us to parametrize the conjugacy classes in $GL(n)$ over any commutative field.
How can we parametriize the conjugacy classes in $SL_n(k)$, where $k$ is an ...
- 11.2k
6
votes
3
answers
482
views
Linear subspaces in cones over orthogonal groups
Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...
- 31.2k
10
votes
1
answer
2k
views
Who first proved that the dimension of a vector space is unique?
every vector space is known to have a basis (assuming the axiom of choice). This is attributed to Georg Hamel (http://de.wikipedia.org/wiki/Georg_Hamel). Moreover, any two bases have the same ...
- 558
0
votes
2
answers
148
views
Bounding 2nd Eigenvalue of a Pseudo-Rotation-ish matrix
Let $p,q$ be arbitrary primes.
Let $N = p * q$.
Let $I$ be the $N * N$ identity matrix.
Let $R$ be the $N * N$ matrix defined as follows:
$R[x_0 * p + y_0, x_1 * p + y_1]=1$ if and only if $x_0+1 ...
- 45
5
votes
1
answer
4k
views
Determinant of a sum of two matrices (one dominating the other)
Let $A$ and $B$ be two $n \times n$ real matrices such that:
$\forall i, j: a_{ij} \geq 0, b_{ij} \geq 0$
let $a_\max$ be the largest entry of $A$ and $b_\min$ be the smallest nonzero entry of $B$; ...
- 128
6
votes
0
answers
465
views
Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
3
votes
0
answers
220
views
Could SVD be used to optimize the partial inner-products?
Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...
- 131
1
vote
0
answers
215
views
Characterizing symplectic matrices relative to a partial Iwasawa decomposition
Fixing notation: for matrices $A,X$ we let $A[X]$ denote ${}^tXAX$.
Let $P_n$ denote the collection of real $n\times n$ positive definite symmetric matrices.
For $Y\in P_n$ we have the usual ...
- 2,274
1
vote
1
answer
298
views
How many zero-constraints can be added to a subspace-restricted matrix before no solution exists?
I'm trying to develop an estimator for the concentration matrix of a Gaussian Graphical Model. I've become stuck in trying to find conditions for the estimator to exist. I have a sufficient ...
- 11
2
votes
2
answers
1k
views
Positive semidefinite decomposition, Laplacian eigenvalues, and the oriented incidence matrix
Suppose $A\in\mathbb{C}^{n\times n}$ is Hermitian and positive semidefinite with some decomposition $A=BB^*$, where $B=(b_{ij})\in\mathbb{C}^{n\times m}$ (not necessarily the Cholesky decomposition). ...
- 475
9
votes
2
answers
1k
views
polynomials with minimal $L_\infty$ norm on multiple disjoint intervals
It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
- 223
1
vote
2
answers
198
views
Finding a vector in the subspace
Given a $d$ dimensional vector $\bar{x} = [x_0,...,x_d]^t$,
how do I minimize $||\bar{x}-\bar{y}||_p$ such that $A\bar{y}=0$, for $p= 0$
i.e.,Minimize $L_0$ norm.
I also have the constraints that $...
- 13
1
vote
2
answers
734
views
Singular matrices with integer entries
I am motivated by the following paper by Greg Martin and Erick B. Wong:
http://www.math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf
Here the authors prove that assuming that the entries of an $n \...
- 26.9k
11
votes
1
answer
838
views
Are these abelian groups free?
Suppose we have a countable, torsion-free abelian group $A$ with the property that for each element $a\neq 0$ the set $D_a=\{x\in A|\exists n\in \mathbb{Z}:nx=a\}$ is finite.
Is $A$ already a free ...
- 11.1k
4
votes
1
answer
484
views
Smith Normal Form
Let $R=Z[x_{1},x_{2},\dots,x_{n}]$ be a multivariate polynomial ring. Is it possible to define a normal form for a general $m \times m$ matrix $M$ with entries from $R$?
- 13.9k
3
votes
0
answers
1k
views
How many iterations are required for the Lanczos algorithm to converge?
I am trying to find the n smallest eigenvalues and eigenvectors of a NxN SPD matrix using Lanczos method. What is the number of iterations usually required? I mean, does it scale as $O(N)$ or $O(\sqrt{...
- 31
3
votes
2
answers
3k
views
How does the Laplace Transform work for circuit analysis? [closed]
I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to ...
- 49
2
votes
1
answer
124
views
uniform bit generator
I'm trying this question for days but no luck, if someone can give me a lead or an article that solves this.. it would be great:
A uniform bit generator is a function $f:\{0,1\}^n \times \{0,1\}^n \to ...
- 21
2
votes
1
answer
446
views
Example for pairwise triangularizable but not all three.
I am not able give an example for the following problem on simultaneous triangularization. So, I thought I will post it here.
Give an example of three linear transformations $A,B$ and $C,$ such ...
- 2,239
2
votes
1
answer
196
views
Relations between a set of inner products of vectors
Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A_1\rangle,\dots,|A_n\rangle$, $\langle A_i|A_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products $\...
- 23
4
votes
4
answers
3k
views
The multiplicity of the max eigenvalue in matrix multiplication
Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \...
- 41
7
votes
0
answers
512
views
maximal subgroups of $GL_2(Z/p^kZ)$
Hello,
is there any classification of proper maximal subroups of $GL_2(\mathbb{Z}/p^k\mathbb{Z})$ for $k>1$ (analogous to the one which exist for $GL_2(\mathbb{Z}/p\mathbb{Z})$)?
Could you give ...
3
votes
2
answers
4k
views
Checking consistency of a system of linear equations and inequalities
I have a lot of systems of equations and inequalities of the following form:
$$ a_{1,1}x+a_{1,2}y+a_{1,3}z+a_{1,4}w = 2 $$
$$ \ldots $$
$$ 0 < x < 2 $$
$$ 0 < y < 2 $$
$$ 0 < z < 2 ...
- 563
6
votes
2
answers
470
views
Alternating multilinear invariants of GL(n) on End (k^n)
Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...
- 33.8k
9
votes
1
answer
2k
views
An iterated tensor product integral
In "Differential equations driven by rough paths" (Terry Lyons, et al) section 1.4.2 it's claimed that the symmetric part of the tensor:
$\int_{0 \le u_1 \le \cdots \le u_j \le t} \mathrm{d}X_{u_1} \...
- 4,304
2
votes
1
answer
586
views
quadratic form factorization
For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p_1x+p_2y+p_3z)(q_1x+q_2y+q_3z)$, are there ...
- 95
6
votes
1
answer
643
views
q-analog of the matrix exponential
I am a fan of the Matrix exponential $\exp(X)$, defined for any complex matrix $X$ by
\begin{equation*}
\exp(X) := \sum_{k \ge 0} \frac{X^k}{k!}.
\end{equation*}
I have a fleeting acquaintance with ...
- 28.6k
4
votes
1
answer
386
views
Given $\mathbf{x}_i^\top A\mathbf{x}_i$ for a SPD matrix $A$ and orthonormal bases $\mathbf{x}_i$, what is the bound of its eigenvalues?
Assume that $A_{d\times d}$ is a symmetric positive semi-definite matrix, and $\{\mathbf{x}_1,\ldots,\mathbf{x}_d\}$ composes a group of orthogonal bases of $\mathbb{R}^d$ where $\mathbf{x}_i\bot\...
- 607
10
votes
4
answers
4k
views
Sum of Gaussian binomial coefficients.
We all know that $\sum_{i=0}^{n}{n \choose i}=2^{n}$. Is there a similar result regarding the q-binomial coefficients? (a.k.a Gaussian binomial coefficients) - $\sum_{i=0}^{n}{n \choose i}_{q}=?$
- 119
2
votes
2
answers
337
views
How do you tell if the span of a set of vectors enters the most positive sector of a graph?
I have $k$ linearly independent vectors in $\mathbb{R}^n$. I want to know if the span of these vectors (i.e. the set of points in $\mathbb{R}^n$ that can be described by linear combinations of these ...
- 693
8
votes
0
answers
738
views
Bounding sum of first singular values squared for Kronecker sum of traceless matrices
Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
- 1,612
13
votes
1
answer
732
views
What is the "positive part" of the unit ball in $M_n(R)$ ?
In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...
- 52.3k
6
votes
1
answer
737
views
Rank of the absolute-value matrix $|M|$ vs. rank of $M$
Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation).
Let $|M|$ be the matrix obtained by taking the absolute value of each entry ...
- 20.2k
0
votes
1
answer
365
views
How to estimate the norm of a matrix
There is a matrix as following,
\begin{eqnarray}
A = \left (
\begin{array}{l}
0 \quad \quad \quad \quad \quad \quad \quad ~~ 1\\
b \quad ~~~0 \quad \quad \quad \quad \quad a\\
ab \quad ~~ b \...
2
votes
2
answers
390
views
Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ?
Consider system of linear equations Ax=0 over $F_2$ (field with two elements {0,1}).
Where number of variables is bigger than equations - so we have many solutions $x$.
Question How to estimate ...
- 24.9k
1
vote
0
answers
165
views
Affine space partition of a general set
Olof Heden, in his work "A survey of the different types of vector space
partitions", discusses various results regarding the following qustion - given a vector space $V$ over a finite field, how can ...
- 119
6
votes
1
answer
180
views
Orbits of exterior products
In linear algebra one learns a lot of normal forms (Which I want to think of as a classification of the orbits of a group action on some set). For example if $V$ is a $k$-vectorspace $GL(V)$ acts on $...
- 11.1k
1
vote
1
answer
2k
views
Recovering a Matrix After Multiplication By Its Transpose [closed]
Given an arbitrary symmetric N-by-N matrix A, how can its original values be calculated from $P$?
$$ P = A'A$$
Both $A$ and $P$ have \( \frac{N^2-N}{2}+N \) degrees of freedom.
Edit: added the ...
- 119
9
votes
1
answer
1k
views
0 eigenvalue for a symmetric tridiagonal matrix
Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...
- 143
4
votes
1
answer
336
views
What is the geometry of the intersection of some cones defined by generalized inequalities?
Hello, considering that for real numbers, the intersection of intervals defined by simple inequalities has a quite simple form as
$$
\bigcap_i\{x|x\leq a_i\}=\{x|x\leq\min_i\{a_i\}\}
$$
However, what ...
- 607
3
votes
1
answer
1k
views
What is the minimum of the Frobenius norm in the intersection of positive semidefinite cones?
For scalar variables $x$, we have a simple solution for the following problem.
\begin{eqnarray}
\min_x&&\alpha(x-a)^2+\beta(x-b)^2 \\\
\mathrm{s.t. }&&x\leq a\\\
&&...
- 607
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{...
- 7,347
1
vote
1
answer
794
views
Tensor Products, Sub-Algebras, Sub-Modules, and Inclusions
Let $A$ be a not neccessarily commutative algebra, and let $B \subset A$ be a subalgebra of $A$. Moreover, let $M$ be an $A$-bimodule, and let $N \subset M$ be a $B$-sub-bimodule. The tensor product $...
- 389
2
votes
1
answer
281
views
Indecomposable extensions of regular simple modules by preprojectives
Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$.
In ...
- 853
5
votes
1
answer
540
views
Cosets of groups of functions
Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$.
The set $\mathcal ...
- 4,312