All Questions
6,289 questions
1
vote
1
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338
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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 ...
- 1,327
12
votes
3
answers
1k
views
Eigencircles of n x n matrices?
An eigenvalue of a 2 x 2 matrix satisfies the equation
$$ \left(\begin{array}{cc} a & b \\ c & d \end{array} \right)\left( \begin{array}{c} x \\ y \end{array}\right) = \lambda \left( \...
- 22.8k
4
votes
1
answer
633
views
ODE in symmetric definite positive matrices
It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt a-\...
- 427
1
vote
2
answers
508
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Sufficient conditions for inverse-positivity
I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the ...
- 6,962
18
votes
2
answers
5k
views
Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
- 345
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 ...
- 6,962
0
votes
0
answers
395
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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} ...
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 $...
- 389
1
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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 ...
- 2,074
8
votes
2
answers
2k
views
Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
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),
$$
where $M$ is fixed, and $J$ is a diagonal matrix (with
0/1 on the diagonal, if it helps.) In my problem
$M$ is fixed,...
- 650
25
votes
3
answers
3k
views
Understanding zeta function regularization
I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced zeta function regularization in a compelling way. The goal is to define the determinant of a positive self-...
- 22.1k
0
votes
1
answer
672
views
Improving on trivial determinant estimates
Given an $n \times n$ matrix $(c_{ij})$ with entries in $\mathbb{R}$ and such that $c_{ij} \leq B$ for some $B > 0$, then it is obvious that the determinant, which we call $\Delta$, is at most $B^...
- 26.9k
3
votes
1
answer
135
views
Mapping a subset of semi-definite matrices through arcsinus
Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ...
- 1,352
5
votes
2
answers
4k
views
sparsity of QR decomposition
Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
- 51
1
vote
1
answer
165
views
Matrix elimination
$A$ is symmetric positive definite matrix and $S$ is such that $A=SS^{T}$. Further
$y=Sz$
Does there exist a simple ( or any verifiable) relation exist only involving $A$,$y$ and $z$ ?
Thanks
- 489
3
votes
1
answer
211
views
Generalizing the spectral radius of a unistochastic matrix
Consider a square matrix $A$, and from it construct $B$ whose entries are the squared magnitudes of those in $A$. What can we say about the spectral radius of $B$? I know that for a unitary matrix $A$,...
- 757
1
vote
0
answers
229
views
Counting equivalence classes in the transitive closure of two equivalence relations
Let $X$ be a finite set, and let $P_i$ and $Q_j$ be two partitions of $X$:
$$\bigsqcup_i P_i = \bigsqcup_j Q_j = X.$$
The finest partition which is nevertheless coarser than both $P$ and $Q$ is ...
- 5,898
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
2
votes
1
answer
271
views
Solvability of the quaternion equations of the Quaternion rationals
So, Let K be the non-commutative field or division ring of the Quaternions. We will consider a sub-field of it L=Q[i,j] in other words, L=Q+Qi+Qj+Qk. Now let us consider the ring of skew polynomials ...
3
votes
1
answer
4k
views
Symmetric unitary matrices
Hi,
Given a complex unitary matrix $U$, can we find a real orthogonal matrix $K$ such that the product $KU$ is a complex symmetric matrix.
Thanks,
- 225
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 ...
- 199
6
votes
1
answer
761
views
Checking if one polytope is contained in another
I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.
At the moment I am ...
- 491
4
votes
2
answers
977
views
Articles with examples of Darboux functions without fixed points
A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \...
- 2,222
9
votes
3
answers
2k
views
Tensor product of linear mappings versus chain complexes
A chain complex of vector spaces $X_k$ is a sequence of linear mappings
$\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} \dots$...
- 5,327
3
votes
1
answer
290
views
Bandwidth reduction of multiple matrices
Suppose I have a symmetric matrix $A$ and several diagonal matrices $D_1,D_2,\dots$. Are there any matrix transformations such as $P^\top A P$ so that
$$P^\top A P, P^\top D_1 P, P^\top D_2 P, \...
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 ...
- 2,829
26
votes
2
answers
3k
views
Sizes of bases of vector spaces without the axiom of choice
Assuming the axiom of choice does not hold we have that there is a vector space without a basis. The situation can be, in some sense, worse. It is consistent that there are vector spaces that have two ...
- 39.8k
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
5
votes
0
answers
275
views
stochastic control / geometric mean
Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
- 427
4
votes
1
answer
1k
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Generalizing inequality relating Euclidean distance & Frobenius norm to Bregman divergences such as relative entropy & von Neumann divergence
Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
- 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.
- 11
16
votes
3
answers
15k
views
Interesting relationships between Cholesky decomposition and diagonalization
Let $\Sigma$ be a hermitian positive definite matrix and $L$ be its Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ is ...
- 1,902
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 ...
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 ...
- 427
2
votes
1
answer
137
views
Design constraint systems over the reals
This question is inspired by the discussion at this problem.
Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...
- 30.1k
0
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1
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269
views
Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex?
Hello, everyone!
As we know that by Jensen's inequality, for jointly convex function $f$ and $\sum_ix_i^2=1$, we have
$$f(\sum_i{x_i^2\lambda_i},\sum_i{x_i^2\theta_i)}\leq\sum_i{x_i^2f(\lambda_i,\...
- 607
3
votes
0
answers
617
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normal form of antisymmetric matrices under pseudo-orthogonal transformations
It is well-known that any real anti-symmetric $n \times n$ matrix $A$ can be transformed via
$A \to O A O^T$ into block-diagonal form consisting of $2 \times 2$ antisymmetric matrices,
where $O \in SO(...
0
votes
0
answers
2k
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In a network with N nodes, what is the general formula for computing the propagation of a set of numbers?
I am creating a circular neural network with N nodes. Each node is connected via a send pathway to every other node, and the connection between two nodes has a weight. Any number sent over the ...
- 1
10
votes
2
answers
3k
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How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
- 693
5
votes
2
answers
562
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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
4
votes
2
answers
1k
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signs of eigenvalues of quadratic form
Let $A=(a_{ij})_{i,j=1}^n$ be a symmetric real matrix, $M_k:=det(a_{ij})_{1\leq i,j\leq k}$ be its minors and $M_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some permutation)...
- 109k
6
votes
1
answer
13k
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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 ...
- 61
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 ...
- 51
5
votes
1
answer
1k
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Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
1
vote
3
answers
585
views
Checking for invertibility of large matrices in MAGMA
If you have a number of large matrices, and you wish to determine whether each matrix has determinant zero or not, what is the most efficient way to do this in MAGMA
(it appears that calculating the ...
- 295
5
votes
2
answers
259
views
Interlacing for "Almost Hermitian" matrices
I am wondering if there is something known about the interlacing properties of an "Almost Hermitian" matrix, in the following sense: let A be a nxn matrix so that it has a Hermitian principal minor of ...
- 6,962
1
vote
2
answers
3k
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Low-rank factorization of SPD matrix
I have a symmetric positive definite (SPD) matrix $A$ that needs to be factorized as ${A=SS^{T}}$. However, using the Cholesky decomposition for this purpose is prohibitive in terms of computational ...
- 489
2
votes
1
answer
232
views
An innocent looking subgroup of $U(n)$
Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...
- 4,466