All Questions
6,290 questions
3
votes
1
answer
204
views
Effects of unitarian multiplication into the spectrum of a finite matrix.
I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$?
This matrix "square-root" has ...
0
votes
2
answers
818
views
a question about the Jordan form [closed]
Some reference say that if rank($A$)=rank($A^2$),then the geometric and algebraic multiplicities of the eigenvalues $\lambda=0$ are equal;that is,all the Jordan blocks correspondint to $\lambda=0$ (if ...
- 139
1
vote
3
answers
5k
views
Number of parameters needed to specify a Hermitian matrix of rank r.
Hi,
i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.
1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank $...
- 199
2
votes
3
answers
924
views
1 or -1 as an eigenvalue of graph
I have a regular and arc transitive graph which I think that either 1 or -1 is an eigenvalues of adjacency matrix of this graph. How can I prove it? Is there any classification of graphs which have 1 ...
- 461
4
votes
2
answers
2k
views
How to efficiently compute the generalized cross product?
It's possible to extend the well known cross product between two vectors in $\mathbb{R}^3$ to $n-1$ vectors in $\mathbb{R}^n$.
Let $\vec{v_1}, \vec{v_2}, \dots, \vec{v}_{n-1} \in \mathbb{R}^n$ and $\...
- 143
0
votes
1
answer
2k
views
Finding linearly independent columns of a large sparse rectangular matrix
I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not ...
- 103
3
votes
1
answer
3k
views
Nonlinear matrix equation
Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw=\lambda_1v+\lambda_2w$
$Aww^TAv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \lambda_3$ are ...
- 41
9
votes
1
answer
808
views
Trace of a functor (or dimension of a category) in extended 2d TQFTs
In an extended 2d TQFT $Z$, a point (with orientation + or -) is assigned a category $Z(+)$ or $Z(-)$. This category should be as close to a vector space as possible: $\mathbb{C}$-linear, monoidal, ...
- 543
5
votes
2
answers
2k
views
Bounding the minimal maximum norm of a solution of a linear system.
I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
- 191
0
votes
3
answers
1k
views
Convex Combination of 2 hermitian matrices
Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices ...
- 1,421
5
votes
1
answer
700
views
Matrices that are > 1 in a sense
How can I characterize the class of square matrices such that:
$ ||MN||_F \ge ||M||_F $?
In other words, when multiplied, they always give "bigger" products.
The norm is the Frobenius norm, which is ...
- 558
3
votes
2
answers
490
views
Constructing equivalent algebraic expressions for matrix equations
I have an expression involving matrices, of the form:
$$f(k)=x^T A_k^{-1}A x$$
where $x$ is a $1\times N$ vector, $A_k = A + k I$ and $A$ is an $N\times N$ matrix ($A_k$ is invertible for all $k$) ...
- 31
2
votes
0
answers
265
views
Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
- 21
6
votes
2
answers
3k
views
On the representation of a (real) square matrix as a product of two symmetric matrices
(For this question, all matrices are real).
According to the ancient paper "Über die Darstellbarkeit einer Matrix als Produkt yon
zwei symmetrischen Matrizen, als Produkt yon zwei
alternierenden ...
- 2,600
1
vote
1
answer
154
views
Optimal weights for large eigenvalues of Laplacian
For a weighted and directed graph $G$ on $n$ vertices we define the Laplacian matrix by $L(G) = D(G)-A(G)$. Here $(i,j)$-th entry of ${A(G)}$ equals the weight $w_{ij}$ of the edge from $i$ to $j$ if ...
- 397
1
vote
0
answers
475
views
How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...
- 1,444
0
votes
1
answer
276
views
Positive definite Hermitian matrices of countable rank
Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the ...
user2529
2
votes
0
answers
1k
views
Eigenvalue problem for symmetric block tridiagonal matrices?
Is there a procedure to find the eigenvalues of $\textbf{M}$?
$$\begin{eqnarray}
\textbf{M}=\left[
\begin {array}{ccccc}
\textbf{A} & \textbf{B} & & &\\
\...
- 21
0
votes
1
answer
163
views
Nonsingular zeroes are algebraic?
I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields):
Suppose $Q\in L^n$, $f_1, \...
- 1,979
11
votes
1
answer
2k
views
Quantifying the failure of the Cholesky factorization test for indefinite matrices
The Cholesky factorization is the classic test to check if a matrix is positive definite. In infinite precision it is also an exact test: A matrix has a Cholesky factorization iff it is positive ...
- 3,217
3
votes
1
answer
2k
views
Rational subspaces
In $\mathbb{R}^n$, we say that a linear subspace is rational if it admits a basis in $\mathbb{Q}^n$ (or equivalently in $\mathbb{Z}^n$). This means that $E\cap \mathbb{Z}^n$ is a submodule of $\mathbb{...
- 830
7
votes
1
answer
975
views
convex hull of pairs of matrices
Is there a simple description of the the convex hull of all the pairs of $n$ by $n$ matrices $(A,B)$ such that $$AA^t+BB^t=A^tA+B^tB=I$$ This is a convex set in dimension $2n^2$, and I am hoping for ...
- 123
3
votes
1
answer
767
views
Linear algebra of finite abelian groups
If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
- 133
8
votes
2
answers
692
views
Smith normal form of a Matrix with -1 outside the diagonal
I am given a matrix of the following form:
$$M = \begin{pmatrix}
a_0 & -1 & \cdots & \cdots & -1 \newline
-1 & a_1 & \ddots & & \vdots \...
- 345
3
votes
2
answers
5k
views
Decomposition of Matrices in Semisimple and Nilpotent Parts
I asked this question in https://math.stackexchange.com/questions/204115/decomposition-of-matrices-in-semisimple-and-nilpotent-parts but remains unanswered.
For any matrix $A\in M_n(\mathbb F)$, ...
- 545
20
votes
2
answers
1k
views
Spectral radius on 0-1 vectors.
Let $A$ be an $n\times n$ symmetric substochastic matrix (i.e. all entries are non-negative and each row adds up to $1$ or less).
Call a vector $v \in \mathbb{R}^n$ an indicator if $v \neq 0$ and ...
- 4,304
2
votes
1
answer
714
views
Is there a natural distance between skew hermitian matrices?
Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to n-...
- 103
4
votes
0
answers
3k
views
Intersection of subspaces
If you have two linear subspaces $V_1$ and $V_2$ of a vector space $V,$ both given by their bases, there is fairly heavy handed way of computing their intersection: write down the projection matrices ...
- 96.4k
11
votes
2
answers
797
views
Three half circles on the plane may not meet nicely
Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...
- 2,136
3
votes
2
answers
2k
views
are intersections of kernels also kernels? [closed]
Suppose $S,T$ are two linear transformations between some fixed pair $V, V'$ of finite-dimensional real linear vector spaces. Now suppose further that $S,T$ have nontrivial kernels in $V$ and that the ...
- 2,274
4
votes
1
answer
393
views
Can an ellipsoid be moved freely inside another ellipsoid?
An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...
- 87
2
votes
1
answer
719
views
Lower bound on Bhattacharya distance between independent Gaussian distributions ?
I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions $p_1\sim\...
- 163
1
vote
0
answers
109
views
Is there a Krylov subspace method for solving D+epsilon*S where D is diagonal, epsilon small and S skew-symmetric
I'm working on a problem that gives a matrix system of the form D + epsilon*S, where S is a skew-symmetric matrix. I'm interested in finding if any work has been done to develop a conjugate gradient ...
- 11
0
votes
2
answers
1k
views
Similarity about unitary matrices
Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in GL_n(\mathbb{C})...
- 651
4
votes
2
answers
543
views
Prove log of eigenvalues are dense in R?
Suppose you have the set of all possible $n$ x $n$ square adjacency matrices where $n$={1,2,3,4...}. For each matrix, compute the logarithm of the largest eigenvalue. Is it true that the set of ...
- 63
1
vote
1
answer
253
views
a variation on the theory of equitable partitions for graphs
Assume you have a graph with an equitable partition with respect to cells $V_1,\ldots,V_n$. Accordingly, you can take the cellwise average value of a function on the node set - in other words, the ...
- 3,388
11
votes
1
answer
265
views
What is the order of the largest subset of M_n(Z_p) such that no two elements commute?
Let $A(n,p)$ be the order of the largest subset of $M_n(Z_p)$ such that no two distinct matrices in this subset commute. Is it true that $\lim_{p \to \infty} \dfrac{A(n,p)}{p^{n^2}} =1$? Can anyone ...
- 413
18
votes
6
answers
6k
views
Computing signature
I have a feeling that this might have already been asked, but can't find the question. Anyway, the question is: given a symmetric $n\times n$ matrix, is there a faster way to compute its signature ...
- 96.4k
2
votes
3
answers
348
views
if Y-X is positive semi-definite, are the eigenvalues of Y bigger?
So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha_1\leq\...
- 87
8
votes
3
answers
1k
views
Eigenvalues of a special block matrix associated with strongly connected graph
Definition
Let $G=(V,E,A)$ be a strongly connected directed graph, where $V=\{1,2,...,n\}$ denotes the vertex set, $E$ is the edge set, and $A$ is the associated adjacency matrix with $0-1$ weighting,...
- 418
8
votes
2
answers
15k
views
Upper bounds on eigenvalues of PSD matrix?
Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums?
For instance, the Perron–Frobenius ...
- 301
3
votes
0
answers
221
views
Eigenvalues vs.matrix sparsity
For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity?
Is there a closed-form expression that states this ...
- 31
20
votes
2
answers
1k
views
a determinantal identity
Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...
- 340
1
vote
1
answer
160
views
Morse index and permutation of diagonal entries of a symmetric matrix
Do there exist results concerning preservation or not of the Morse index of a symmetric matrix $A$, after permuting its diagonal entries, and keeping fixed the off--diagonal ones?
Thanks!
- 13
10
votes
1
answer
813
views
Linear system of equations with nonnegative solutions and a recursion rule
My question derives from reading a recent preprint (arXiv:1209.0827v1, in particular Section 4.1), but it can be phrased quite independently from that paper. The setup is as follows.
Let $A$ be the ...
2
votes
1
answer
158
views
Destroying the structure of a linear system while preserving its maximum eigenvalue
I have an asymmetric square matrix with non-negative real entries in the range [0,10], representing the edge-weights of a directed network. Assume that the network is a linear system. My general ...
- 21
3
votes
1
answer
201
views
Concavity of Spectral mean
The geometric mean of two positive definite matrices $A, B$ is defined by $A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$. The following inequality holds true $$\left(\sum_{i=1}^n A_i\right)\sharp ...
- 478
12
votes
5
answers
9k
views
Solving Lyapunov-like equation
The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
- 121
5
votes
5
answers
1k
views
Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid
A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...
- 30.3k
3
votes
0
answers
103
views
Finding lattice vector with entries of low height
Hello
Given a lattice $L \subseteq \mathbb{Q}^n$ we can define the height of a given element $v = (v_1,\ldots,v_n)\in L$ to be
$$ \operatorname{ht}(v) = \sum_{i=1}^n \operatorname{ht}(v_i)$$
where ...
- 71