All Questions
6,178 questions
6
votes
0
answers
514
views
concentration for eigenvectors
I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
- 6,962
6
votes
1
answer
13k
views
what are the conditions for the product of 2 symmetric matrices being symmetric [closed]
In generally, the product of two symmetric matrices is not symmetric, so I am wondering under what conditions the product is symmetric.
Likewise, over complex space, what are the conditions for the ...
CommunityBot
- 1
0
votes
2
answers
194
views
Matrices whose range is equal to the column set [closed]
Is there such a thing?
I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).
- 148k
0
votes
1
answer
148
views
Augmenting $t$-dimensional sub-spaces into $t+1$-dimensional sub-spaces through a basis
Let $V_1, V_2, ..., V_n$ be $t$-dimensional sub-spaces of an $n$-dimensional vector space $V$
where $t \lt n$.
Under what conditions the following would be true:
for any $B= \{v_1, v_2, ..., v_n\...
- 1,034
13
votes
1
answer
702
views
Integer matrices with a strange divisibility property
Fix an integer $n$. What can you say about a (not necessarily square) matrix $A$ with integer entries that has the property that for any $k$, every $k\times k$ minor of $A$ is divisible by $n^{k-1}$? ...
- 22.1k
4
votes
3
answers
4k
views
upper bounds on a certain matrix norm
Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?
- 54.2k
3
votes
1
answer
471
views
from affine matroid to measures
Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let
$W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional
simplices (i.e. bases of the ...
CommunityBot
- 1
3
votes
1
answer
447
views
Does a product of matrices have eigenvalue 1
Start by fixing invertible matrics $A_1, \ldots, A_m \in \mathbb{Z}^{n \times n}$.
For a sequence $i_1, \ldots, i_k$ we construct $A = A_{i_1} \cdots A_{i_k}$. We would like to know "Is 1 an ...
CommunityBot
- 1
7
votes
3
answers
2k
views
Canonical form of symmetric integer matrix M
Let $M$, $N$ be a symmetric matrix over a ring $R$.
$M$ and $N$ are said to be equivalent if there exist an invertible
matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U^T$ is the transpose ...
- 25.7k
13
votes
3
answers
2k
views
Which polynomials are determinants of a symmetric matrix with linear entries?
Let $k$ be a field. Can each degree $n$ polynomial $P(t) \in k[t]$ be written as the determinant of the matrix $A + tB$, where $A$ and $B$ are two symmetric $(n \times n)$-matrices with entries in $k$?...
- 4,279
1
vote
2
answers
271
views
small sums of entries in submatrices - strange phenomenon
Suppose that $x \in \mathbb{R}^{n}$ is a vector of small positive fractions, i.e. $x_{i} \approx \frac{1}{n}$. The exact values are unknown. I form the matrix $M=diag(x)-\frac{xx^{T}}{2}$ which is a ...
- 37.7k
3
votes
1
answer
473
views
Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)...
Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...
CommunityBot
- 1
2
votes
1
answer
815
views
A question for solutions of perturbed linear systems
Consider a linear system
$$Ax=b\qquad (*)$$
and a sequence of perturbed linear systems $$(A+\delta A_n)x=b+\delta b_n. \qquad (n)$$
Suppose that all the linear systems are consistent (i.e., ...
CommunityBot
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8
votes
1
answer
2k
views
Complexity of finding a 0-1 vector in a subspace or showing that there is none
This question, is a slightly different disguise (see below), came up in discussions of this question about equitable partitions
A $0,1$ vector in $\mathbb{Z}^n$ is any vector with all entries $0$ and ...
CommunityBot
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6
votes
2
answers
421
views
Triangularizing a matrix with function entries
Hi Everybody!
Given a matrix, with smooth functions as arguments is there any result which say about its triangularization?
I know that, the question is in affirmative for diagonalizing a matrix ...
- 2,239
4
votes
1
answer
461
views
smooth dependency of eigenvalues of parametrized unitary matrice
Hi all. I recently encountered a problem as following: if you are given a family of unitary matrices $A(t)$, in which $t\in\mathbb{R}$ is the parameter and especially $A(t)$ is smooth in $t$, which ...
- 798
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). ...
CommunityBot
- 1
3
votes
0
answers
130
views
Computing the norm of the columns of an implicitly defined matrix
I have an $n \times n$ matrix $M = \Sigma W$ where $\Sigma$ is diagonal and $W$ orthogonal. $W$ is implicitly defined, i.e. I can only perform matrix-vector products (but I also have access to $W^T$).
...
1
vote
0
answers
212
views
componentwise eigenvector perturbation
Does the sin-theta theorem imply a componentwise nonasymptotic bound for eigenvectors? Assume, for the purpose of this question, that the eigenvalues concerned are simple.
If this is trivial, I ...
- 6,962
0
votes
0
answers
276
views
Another matrix diagonalization problem
Given the matrices $X$ and $Y$ in $[0,1]^{n\times m}$, for $n > m > 3$, so that $X1_m=1_m$ and $Y1_m=1_m$, where $1_m$ denotes a $m$-length column vector of ones, find a matrix $Q$ in $R^{m\...
- 51
5
votes
2
answers
562
views
Perron Frobenius with one negative pair of entries
Suppose you have a real symmetric matrix $A$ which is positive except for $a_{ij},a_{ji}$, who are negative.
While it is not generally true that the eigenvector of the dominant eigenvalue of $A$ is ...
- 6,962
1
vote
2
answers
575
views
matrix stability criterion
I have a $5 \times 5$ parametric nonnegative matrix and want to show that it's stable (in the sense that all eigenvalues are positive). It is not symmetric, but I do know in advance that it has 5 real ...
- 6,962
2
votes
1
answer
297
views
equations over (some) lie groups
To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the ...
CommunityBot
- 1
0
votes
3
answers
302
views
Vector "product" diagonalization
Consider a vector space $V$ (with dimension $n+1$ and elements $v$) on which a (commutative and associative) "product" $\odot$ taking $V\odot V\rightarrow V$ is defined, and an $1$ element $v_0$ ...
- 148k
1
vote
3
answers
202
views
Solving for an operator by minimization
Please note that I am looking for numerical algorithms that will tell me what the operator is that minimizes a problem.
I have a 2x2 complex hermitian operator that is a function of two variables, so ...
- 773
2
votes
0
answers
240
views
Copositive matrix?
I want to check under what conditions a matrix of the form $\alpha J -Q$ is copositive, where $J$ is the all-ones matrix and $Q$ is doubly nonegative (i.e. entrywise nonnegative and positive ...
- 6,962
15
votes
3
answers
738
views
Finding lots of discrete vectors in fairly general position
How many vectors can there be in $\mathbb{F}_2^{2n}$ such that no $n$ of them form a linearly dependent set? The bounds I have so far are embarrassingly far apart, though that probably means I should ...
CommunityBot
- 1
1
vote
4
answers
741
views
A matrix diagonalization problem
For matrices $X,Y\in [0,1]^{n\times m}$, for n > m, is there a square matrix $W\in R^{n\times n}$ so that $X^TWY$ is diagonal if and only if $Y = X$? Furthermore, $X$ and $Y$ are column normalized so ...
- 39.9k
10
votes
2
answers
2k
views
What is the canonical isomorphism between the tensor products of the top exterior powers associated to exact sequences of vector spaces?
One often reads (and writes) that an exact sequence of finite dimensional vector spaces
$$
0 \rightarrow X_1 \rightarrow X_2 \rightarrow \dots \rightarrow X_n \rightarrow 0
$$
induces a canonical ...
- 34.7k
1
vote
1
answer
338
views
When do the invariant factors of a direct sum of matrices correspond to those of its summands?
(Tried asking this on math stackexchange, but no takers so far.)
I'm trying to prove something about matroids, which I have reduced to the following question:
Suppose I have a matrix $M$ which is a ...
- 52.3k
0
votes
0
answers
395
views
The ratio of two strictly increasing functions
Given:
\begin{equation}
f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c}
K \\\
i \\
\end{array} \right) \left(-1-\frac{1}{ar}\right)^i
\end{equation}
\begin{equation}
f_2(a)=\sum_{i=1}^{k^*-1} ...
CommunityBot
- 1
3
votes
1
answer
1k
views
Explicit formula for Cholesky factorization in a special case
I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a ...
- 3,934
1
vote
1
answer
168
views
Coinvariant Subalgebras of Hopf Comodules and Quotients
For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta_R(W) \subseteq W \otimes H$, and note that this implies that $...
- 47.3k
1
vote
0
answers
182
views
matrix-theoretic terminology query
Is there an accepted term for the following property?
Let $A$ be a real matrix such that all entries of the eigenvector corresponding to the least eigenvalue have the same sign.
NOTES: (1) The case ...
- 6,962
0
votes
2
answers
156
views
inclusions of linear colimits into smooth manifolds
Let $V$ be the category of finite dimensional vector spaces and $M$ the category of
smooth finite dimensional Hausdorff manifolds.
Now suppose any finite dimensional vector space is equipped with a ...
CommunityBot
- 1
3
votes
1
answer
323
views
determinant of diagonal - fixed
I have to study/evaluate many determinants of the form
$$
f_M(J)=\det(J-M),
$$
where $M$ is fixed, and $J$ is a diagonal matrix (with
0/1 on the diagonal, if it helps.) In my problem
$M$ is fixed,...
CommunityBot
- 1
2
votes
2
answers
820
views
eigenvalues of edge regular graphs
In graph theory, an edge regular graph is defined as follows.
Let G = (V,E) be a regular graph with v vertices and degree k.
G is said to be edge regular if there is also integer λ such that:
Every ...
- 6,962
5
votes
5
answers
2k
views
median of matrices
I have $n$ positive definite Hermitian matrices $M_n$ and I want to define and compute their median.
These matrices correspond to independent estimations of a covariance matrix in the presence of ...
- 265
3
votes
1
answer
135
views
Mapping a subset of semi-definite matrices through arcsinus
Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ...
- 1,352
0
votes
1
answer
672
views
Improving on trivial determinant estimates
Given an $n \times n$ matrix $(c_{ij})$ with entries in $\mathbb{R}$ and such that $c_{ij} \leq B$ for some $B > 0$, then it is obvious that the determinant, which we call $\Delta$, is at most $B^...
- 54.2k
5
votes
2
answers
4k
views
sparsity of QR decomposition
Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
- 6,962
0
votes
0
answers
109
views
Expansion (asymptotic) of scalar function of a square matrix , in terms of determinant of argument?
The title says it all. I have a scalar function (really, a determinant) of a square matrix argument. Can I find an (asymptotic) expansion of the function, in a series in the determinant of the ...
- 2,600
1
vote
1
answer
165
views
Matrix elimination
$A$ is symmetric positive definite matrix and $S$ is such that $A=SS^{T}$. Further
$y=Sz$
Does there exist a simple ( or any verifiable) relation exist only involving $A$,$y$ and $z$ ?
Thanks
- 21.5k
6
votes
2
answers
503
views
Unpublished work of Wielandt
Wielandt wrote a paper titled "Remarks on diagonable matrices".
According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis
by Helmut Wielandt, Hans Schneider, Bertram Huppert ...
- 11
2
votes
1
answer
851
views
Null Space Perturbations
Hi,
I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases.
The distilled version of the ...
- 20.2k
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
14
votes
1
answer
1k
views
A Question on Random Matrices
Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by
$$
V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q})
$$
where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
- 3,626
3
votes
0
answers
385
views
Does this inequality of negative relative entropy and quantum relative entropy hold?
Hello, everyone!
Question
I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices $...
- 607
0
votes
1
answer
576
views
Solving Ax=b, where A is an unknown Toeplitz matrix, x and b are known.
I am trying to solve an equation of the form $Ax=b$, where $A$ is an unknown Toeplitz matrix, while $x$ and $b$ are known.
If one knows corresponding Matlab procedure, it'll be great.
- 427
4
votes
0
answers
154
views
connectivity in automata by words of length n-1
Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected?
That is for any pair of distinct ...
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- 1