All Questions
6,292 questions
8
votes
3
answers
1k
views
Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?
Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.
Does an upper ...
- 3,165
7
votes
1
answer
251
views
Solving for a set of points from projections
Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
S = \{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let $v \in \mathbb R^n$ and consider the image set (not counting ...
user45183
3
votes
0
answers
171
views
Generalized family of Hölder inequalities
Is the "only if" direction of the following fact known?
For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^...
- 31
2
votes
1
answer
166
views
Bound on the 2-norm of a "special" matrix
Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and $F\...
- 215
1
vote
2
answers
574
views
when will the surfficient large power of a rational matrix be a integer matrix?
$A$ is a $n\times n$ matrix whose elements are all non-negative rational numbers and $Det(A)$ is a non-zero integer.Under what condition the following is true?(0) There exist a positive integer $M$ ...
- 29
3
votes
1
answer
773
views
General method for under and over determined systems?
Suppose I have a system:
$$
Ax = b
$$
where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.
I'd like a combination of a ...
- 723
5
votes
0
answers
604
views
The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)
(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
- 10.5k
1
vote
0
answers
1k
views
Diagonal entries of a Cholesky factorization
Let $I$ denote an identity matrix, $E$ denote the all-one matrix of dimension $k\times k$ and $c$ some positive real number. Define $X=B(I-cE)B^T$ where $B$ is given by
$B:=\begin{pmatrix}
1 &\...
- 29
-2
votes
1
answer
594
views
Is dual cone unique? [closed]
Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition ...
- 187
4
votes
2
answers
657
views
Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v2 §19)
$\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is ...
- 33.8k
2
votes
0
answers
210
views
Dominant eigenvalue of sum of tridiagonal and diagonal matrices
Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...
- 21
2
votes
1
answer
78
views
An algebraic equation question [closed]
My question is this:
If $\frac{\sqrt[n]{\prod_{i=1}^n(p_i + 1)}}{\sqrt[n]{\prod_{i=1}^n(m_i + 1)}} = e ^\beta$
can I find an expression (either exact or approximate) for $\frac{\sqrt[n]{\prod_{i=1}^...
- 163
2
votes
0
answers
91
views
Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings
In one paper from 1980 I found a note that there are no known algorithms for solving
homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring.
(The non-...
1
vote
1
answer
542
views
Eigendecomposition of a summation of matrices [duplicate]
Can anyone tell me if there's a way to relate the eigendecomposition of the result of a summation of matrices with the eigendecomposition of those matrices?
More specifically:
If I have a matrix $K = \...
- 19
8
votes
1
answer
515
views
Casson invariant
Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does ...
- 96.4k
6
votes
1
answer
661
views
Diagonalization for sums of Hermitian matrices
I found an interesting question about diagonalizable matrices,
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ Hermitian, such that $AB\neq BA$.
Do there exist complex numbers $u\neq v$, such that $A+uB$ and ...
user49093
1
vote
0
answers
517
views
How do I prove a matrix A is self adjoint to an inner product? . [closed]
In the source question
B is an element of $M_n(R)$ and is a symmetic matrix
such that $v^tBv>0$.
Also $<.|.>$ is an inner product on $R^n$ called the $B$-inner product.
we are asked to ...
- 11
4
votes
0
answers
382
views
Efficiently factorize a KKT system with block diagonal upper corner
I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form:
\begin{equation}
A =
\left[\begin{array}{c|c}
\...
- 723
3
votes
1
answer
1k
views
Symplectic block-diagonalization of a complex symmetric matrix
This is a follow-up question to the one asked here:
Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
- 290
1
vote
2
answers
172
views
Linear Programm with matrix [closed]
Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
- 11
-1
votes
1
answer
200
views
Collecting terms of a linear expression with nested sums and combinatorics in coefficients
I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ \sum_{j=2}^{m-1}...
4
votes
1
answer
299
views
variation of the Lieb concavity theorem
A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B:
$$
(A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \}
$$
for $s \in ...
- 483
5
votes
2
answers
1k
views
Lie's theorem in characteristic $p$
Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
- 3,741
3
votes
0
answers
92
views
Injectivity of a linear logistic transform
The motivation for this question has to do with neural networks, but it is essentially a purely mathematical question.
Suppose you have a perceptron with one hidden layer, a bias, and a logistic ...
- 1,902
8
votes
1
answer
364
views
Sets of cardinalities of bases without choice
For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...
- 20.5k
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
3
votes
0
answers
224
views
Find the Range of Function
What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$,
$$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$
where $b_k=\prod_{j\neq k}(z_j-z_k)$ for $1\leq k\leq m$ ?
- 713
1
vote
1
answer
136
views
Any generic way to move a psd matrix to its neighbors?
Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...
- 363
4
votes
1
answer
135
views
Characterization(?) of coersive(?) elements in the special linear group
Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that
$\|A\| < x$ and $\|...
- 96.4k
10
votes
2
answers
631
views
What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$
The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
- 700
2
votes
0
answers
146
views
Lanczos algorithm with thick restart on a dynamic matrix
currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...
- 21
16
votes
1
answer
818
views
Lift chain complex from $\mathbb{F}_2$ to $\mathbb{Z}$
We start with a finite dimensional chain complex over $\mathbb{F}_2$, equipped with a basis. That is, we have finitely many finite dimensional $\mathbb{F}_2$-vector spaces $C_0,\dots,C_k$ with bases $...
- 518
1
vote
0
answers
147
views
Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121
5
votes
1
answer
983
views
About partial uniqueness of SVD
In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let $...
- 153
4
votes
0
answers
101
views
successive schur complements
If I have a large (e.g. 6000x6000), sparse, positive definite matrix $M$ (which may have individual entries everywhere, but most non-zero entries are on / around the diagional).
Divide $M$ into blocks ...
- 41
6
votes
0
answers
489
views
Symmetric matrices with $\rho(A)\gg\|A\|_\infty$
For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
- 23k
2
votes
0
answers
477
views
Norm bound of a complex resolvent
A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...
- 73
6
votes
1
answer
417
views
Simultaneous Tridiagonalization of a given set of hermitian matrices?
I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?
- 1,421
5
votes
3
answers
2k
views
How to check whether a matrix is completely positive or not?
The definition:
cone of completely positive matrices
$$
\mathcal{C}=\left\{
\sum_{i=1}^kx_ix_i^T : \text{$x_i\in\mathbb{R}^n_+$ for $i=1,2,\ldots,k$}
\right\}.
$$
I just don't know how to check ...
- 187
4
votes
1
answer
751
views
submatrix of a given size with maximum frobenius norm
Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
- 859
9
votes
0
answers
434
views
When is a product of hyperbolic matrices hyperbolic?
Suppose $A_1,\ldots,A_n$ is a sequence of $2 \times 2$ complex matrices such that $| \det(A_j) | =1$ and $ | \mathrm{tr}(A_j) | > 2 $ for each $j$. What kinds of reasonable restrictions can one ...
- 437
19
votes
2
answers
9k
views
Distributing the Hodge map over the wedge product
Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to "...
- 191
1
vote
2
answers
220
views
reference needed for sobolev type estimates
I'm reading a paper and the authors applied the following sobolev type estimates
$$
||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}
$$
for $\alpha>\frac{1}{4}$,
where $v$ ...
- 113
0
votes
0
answers
91
views
Complexity of turning a d-degree polynomial to 2-degree polynomial
For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
- 187
86
votes
1
answer
6k
views
Are there non-scalar endomorphisms of the functor $V\mapsto V^{**}/V$?
Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...
- 4,416
3
votes
0
answers
170
views
Is there such a matrix in $SO(n)$?
Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = \frac{B_{ij}}{\sqrt{B_{ii}B_{jj}}},...
- 131
8
votes
4
answers
2k
views
NP-hard problems in linear algebra and real analysis [closed]
I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...
4
votes
3
answers
2k
views
Square root of a complex matrix commuting with a given one
Assume two commuting $n\times n$ complex matrices $A$ and $B$ are given. Then it is in general false that if $C$ is a square root of $A$, i.e., if $C^2=A$, then $C$ commutes with $B$ (the simplest ...
- 6,777
0
votes
2
answers
179
views
Analyticity of Logarithmic Integrals
Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
- 1,583
1
vote
1
answer
686
views
dual space of the quotient space of some locally convex topological space
I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
- 1,835