All Questions
6,292 questions
13
votes
1
answer
516
views
Permanent of a matrix of odd integers
It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...
- 273
0
votes
1
answer
212
views
L a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L).
Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the proof I followed proves that that the ...
17
votes
1
answer
3k
views
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
12
votes
1
answer
1k
views
A generalization of intermediate value theorem on R^k
Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...
- 207
3
votes
2
answers
275
views
Distinct eigenvalues of the quadratic eigenvalue problem
Suppose we have a quadratic eigenvalue problem $(A_{0}+\lambda A_{1}+ \lambda^{2} A_{2})x=0$. I'd to know if there are conditions under which the problem is known to have a small number of distinct ...
- 6,962
34
votes
13
answers
6k
views
Elementary applications of linear algebra over finite fields
I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
9
votes
1
answer
3k
views
Connection between eigenvalues of matrix and its Laplacian.
Hello!
There are two definitions of graph spectrum:
1) Eigenvalues of adjacency matrix $A$.
2) Eigenvalues of Laplacian of adjacency matrix ($L$).
Different sources offer different properties based ...
- 89
5
votes
4
answers
8k
views
Proving a determinant = 0
The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...
- 51
9
votes
2
answers
1k
views
Rescaling positive definite matrices to force a unit eigenvector
Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones.
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$...
- 193
2
votes
1
answer
386
views
Decomposing bilinear forms in Hilbert spaces
You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
- 1,211
38
votes
2
answers
6k
views
Over which fields are symmetric matrices diagonalizable ?
The question is motivated by this one real symmetric matrix has real eigenvalues - elementary proof:
Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed fields) ...
- 567
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
7
votes
2
answers
2k
views
Tails of sums of Weibull random variables
Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
- 1,791
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
2
votes
0
answers
259
views
Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices
How does the spectra of $DU$ change when $D$ runs over all diagonal unitary matrices? Here $U$ is a fixed unitary matrix. Precisely, let spec$(X)$ be a set of eigenvalues of $X$.
For a unitary matrix $...
- 723
-1
votes
2
answers
605
views
Approximating a subspace by sampling a base without replacement
Let $X$ be a $p \times n$ matrix, with $p > n$. Now, suppose I sample $m < n$ columns from $X$ at random, without replacement. I would like to characterize the distance between the subspace ...
- 461
15
votes
3
answers
5k
views
How to show a certain determinant is non-zero
For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that
the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant
where $\lambda_1 \lt \lambda_2 \lt \ldots \...
- 251
4
votes
1
answer
821
views
Is there a relationship between Entropy of a fininte distrete probability distribution and the squre sum of the values of probability mass function of that distribution?
Sorry for the long title. What I mean is that for two vectors (a_1,...,a_n) and (b_1,...,b_n) with the property $a_i,b_i \geq 0 $ and $ \sum a_i =\sum b_i =1$.
If $ -\sum a_ilog(a_i) > -\sum ...
- 87
3
votes
1
answer
4k
views
Schur complement and negative definite matrices
Hello,
My question regards to the Schur complement lemma. Consider the matrix $M=\left( \begin{array}{cc}
A & B\\\
B^T & C \end{array}\right)
$.
According to the lemma $M\geq0$ iff $C>0$ ...
- 155
15
votes
1
answer
858
views
Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
- 34.7k
1
vote
1
answer
206
views
What is such an equation called?
Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?
$Ax+f(\lambda)Bx=g(\lambda)x$.
- 6,962
12
votes
2
answers
3k
views
On the positive definiteness of a linear combination of matrices
In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
- 895
5
votes
1
answer
346
views
Linear maps preserving positive semidefiniteness
I know of Choi's theorem and some related problems, but not a solution to this exact problem:
Characterize the linear maps from the space $S_n$ of symmetric $n \times n $ matrices to itself that ...
- 20.2k
2
votes
1
answer
970
views
Finding a subspace disjoint from a union of subspaces
Let $k$ be a finite field (I care about $\mathbb F_p$, especially $\mathbb F_2$) and let $V_1,...,V_N\subset k^n$ be subspaces.
I want to find a subspace $S\subset k^n$ such that $S\cap V_i=0$ for ...
- 1,147
7
votes
2
answers
518
views
Morse lemma with least amount of regularity.
I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
- 1,211
4
votes
0
answers
70
views
Recovering Shared Eigenvector Set
Suppose we are given a set of $M$ pairs $\{(\vec{x}^{(i)},\vec{y}^{(i)})\}$, with
$\vec{x}^{(i)}\in\mathbb{R}^N$,
$\vec{y}^{(i)}\in\mathbb{R}^N$,
$M\gg N$ such that
$\vec{y}^{(i)} = Q^{(i)} \vec{x}^...
- 1,667
0
votes
1
answer
590
views
Strictly diagonally dominant hermitian matrices eigenvalues sign
Let $A\in \mathcal{M}_{n\times n}(\mathbb{C})$ be a strictly diagonally dominant hermitian matrix.
My main goal is to tell how many positive eingenvalues $A$ has in terms of its leading diagonal ...
- 3
3
votes
1
answer
451
views
Singular values of the sum of A and A^T
As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}(A+A^{T})$ for a stochastic $A$ (as a function of the singular values of $A$), which is generally ...
- 225
26
votes
2
answers
3k
views
Singular values of sequence of growing matrices
I asked this question on math.stackexchange and haven't received an answer in two weeks, so I'm repeating it here.
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \cr
1/2 & 0 &...
- 656
1
vote
0
answers
358
views
Kernel of modified Kronecker sum
The Kronecker sum of two matrices $A \in M(n \times n, \mathbb{R})$ and $B \in M(m \times m,\mathbb{R})$ is defined by the matrix
$$A \oplus B = A \otimes I_m + I_n \otimes B \in M(nm \times nm, \...
- 397
6
votes
2
answers
1k
views
Canonical form of a general Bilinear Form
There is a simple canonical form of a symmetric and antisymmetric bilibear forms. Is there a canonical form for a general bilinear form?
- 61
0
votes
1
answer
156
views
Projection and Positive matrices [closed]
Sorry my question was not clearly stated. I will ask it more clearly.
Let $G$ be a matrix with only nonnegative elements with linearly independent
columns. Then there exists a column, ${\bf g}$ of $G$...
0
votes
1
answer
296
views
Projection and Positive matrices
Sorry my question was not clearly stated. I will ask it more clearly.
Let $G$ be a matrix with only nonnegative elements with linearly independent
columns. Then there exists a column, ${\bf g}$ of $G$...
2
votes
1
answer
284
views
Dimension of An Ultraproduct Field as a Vector Space over Another Ultraproduct Field
Suppose $F \subseteq K$ are fields with $G$ an ultrafilter on an infinite set $X$. If $F^{\ast}$ and $K^{\ast}$ represent the ultraproducts respectively of $F$ and $K$, it is easy to see that $[K : F]...
- 43
4
votes
2
answers
1k
views
Minimum eigenvalue of a Affine Combination of two Hermitian matrices
Consider two $N \times N$ hermitian indefinite matrices $A_1$ and $A_2$. Consider their affine combination
\begin{align}
M(t)=(1-t)A_1+tA_2
\end{align}
I am interested in the minimum eigenvalue of $M(...
- 1,421
0
votes
0
answers
166
views
Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?
Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
- 87
0
votes
1
answer
239
views
Real Pfaffian representations of real cubic surfaces
Consider the following classical construction (which is called Pfaffian representation as Sasha indicates):
Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space
of ...
- 14.3k
3
votes
1
answer
241
views
Counting integer matrices with specific values of minors
Are there any results concerning the following problem:
Count all $n \cdot (n + m)$ integer matrices ($m \ge 1$) with norm ($\|A\| = \max{|a_{ij}|}$ or $\|A\| = \sqrt{\sum{a_{ij}^{2}}}$ ) less than ...
- 123
1
vote
1
answer
600
views
Linear (in)dependence of minors of a matrix
From (Italian, very nice book):"Lezioni di Geometria Analitica e Proiettiva" by Beltrametti, CArletti, Gallarati, Bragadin (pag. 21):
Let $K$ a field, $V:= K^{n+1}$ and let $e_1,\ldots, e_{n+1}$ a ...
- 4,165
1
vote
1
answer
141
views
Numerical optimisation for multivariate Gaussians
Hi,
I want to calculate
$
f_{\mathbf x}(x_1,\ldots,x_k)\, =
\frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}}
\exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}...
- 619
7
votes
0
answers
209
views
Numerical linear algebra: how to compute $B^TC^{−1}B$ efficiently
Hi,
my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$.
The matrix $C$ is huge ($n$ up to a ...
- 619
1
vote
1
answer
4k
views
How to solve this optimization with the orthogonal constraint?
Problem
Supposing that $A$ is a symmetric real matrix and $\{\mathbf{w}\_i\}_{i=1}^n$ is any orthogonal basis on $\mathbb{R}^n$ such that $W^\top W=WW^\top=\mathbf{I}_n$ where $W=\left[\mathbf{w}_1\;\...
- 607
2
votes
2
answers
1k
views
How to solve a system of quadratic equations over finite fields?
Hello!
Let $m$ be an odd positive integer. Let $r$ be the smallest extension order of finite field $GF(q)$ that the $m$-th roots of unity are in $GF(q^r)$. For instance, it can be assumed that $q = 2$...
- 21
3
votes
2
answers
186
views
Triangularizing a function matrix with smooth eigenvlaues
Given a matrix with function entries, which are smooth and homogeneous, and having smooth eigenvalues, can we find a conjugating matrix with smooth and homogeneous entries that triangularize the given ...
- 2,239
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
4
votes
2
answers
344
views
Impossibility of continuously picking k independent rows from a rank k matrix
Suppose I have an $n\times n$ real (or complex) matrix of rank $k$, and I want to pick $k$ linearly independent rows from it. I want to do this in a continuous fashion as the matrix varies ...
- 82.7k
14
votes
2
answers
608
views
Condition number of matrix after partial orthogonalization
I'm wondering about which bounds one can put on the condition number of
a $n\times n$ square matrix which is obtained from another $n\times n$
square matrix by orthogonalizing the first $m < n$ ...
- 141
2
votes
1
answer
480
views
On linear automorphism on positive definite matrices.
I saw a statement in [Murakami, On automorphisms on Siegel domains] that every linear automorphism $\phi$ on the set of positive definite matrices can be represented as conjugation: i.e. there is a ...
- 537
1
vote
1
answer
417
views
Decomposition of Matrix to its sub-matrix with constant rank
When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ...
- 4,784
1
vote
0
answers
126
views
Matrix Minimax problem
I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The $M_k$...
- 4,829