All Questions
6,177 questions
4
votes
2
answers
356
views
Infinite products of representations of the additive group
Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ locally nilpotent if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. ...
- 5,654
1
vote
0
answers
140
views
Diagonalizing matrices of linear forms of indeterminates
Let $B$ be a matrix with elements as linear forms of indeterminates. Is there a proper diagonalization procedure for such matrices like those of matrices with real and complex entries?
- 13.9k
3
votes
2
answers
2k
views
Eigenvalues of sum of an adjacent matrix and a constant
$A$ is an adjacent matrix of a network. $la$ is the largest eigenvalue of $A$ and $Va$ is its corresponding eigenvector.
I am interested in the following martix: $bA+c-dI$ ($b$, $c$, and $d$ are all ...
- 28.6k
1
vote
1
answer
720
views
A question on gauge functions
In the second paragraph on Page 71 of the book Matrix Analysis by
Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 from
...
- 28.6k
0
votes
1
answer
711
views
a system of nonlinear equations (power sum)
greetings . is there a general method-algorithm to solve the following system !?
$\sum_{n=1}^{m} {x_{n}}^{j}= {k}_{j} $
$j=1,2,...,m$
$k_{j}$ are constants
thanks in advance
CommunityBot
- 1
7
votes
1
answer
530
views
Do real vectors attain matrix norms?
I apologize if the following question ends up being too elementary for this website; I asked it on math.SE a week ago and it remains unanswered.
Let $A$ be an $n \times n$ matrix with real entries ...
CommunityBot
- 1
4
votes
1
answer
741
views
Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer
Let $A$ be a noncommutative finitely generated algebra with a finitely generated set of relations. Moreover, assume that $A$ is finite dimensional as a vector space.
What I want to know is, can ...
- 3,078
3
votes
2
answers
2k
views
When can an eigenvector be chosen uniquely which is invariant to permutation?
Suppose $A\in\Re^{n\times n}_{sym}$ is a symmetric matrix with eigenvalues $\lambda_1,\dotsc,\lambda_n$ in decreasing order. What I seek is a way to choose an eigenvector that is invariant to ...
- 148k
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
CommunityBot
- 1
9
votes
2
answers
954
views
Extremal properties of the determinant for matrices with entries in a fixed subset of $[-1,1]^{n^2}$?
Given a multiset $S\subset [-1,1]^{n^2}$, we set
$$m(S)=\min\vert \det(M)\vert$$
where the minimum is over all matrices with entries forming the multiset $S$
and
$$a(n)=\max m(S)$$
where the maximum ...
CommunityBot
- 1
0
votes
1
answer
314
views
intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$
Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\...
- 23.5k
4
votes
1
answer
358
views
Singular vectors of $AB^{\top} $ versus $B^{\top} A$
Let $A$ and $B$ be matrices of dimensions $d \times n$. Let $C = AB^{\top}$.
We also know that $C = I \mathrm{diag}(\gamma) J$ for some matrices $I$ and $J$ and vector $\gamma$ of length $m$, $m < ...
- 41
1
vote
0
answers
298
views
Norm preserving matrix fix
Hello,
I'll state the problem first and than I'll a little bit of motivation.
Lets be given regular matrix $M \in \mathbb{R}^{n\times n}$ and norm $||.||$ in $\mathbb{R}^{n}$. Define $$ U =\{ L\in \...
- 25.7k
1
vote
0
answers
467
views
Is it possible to compute the eigenvalues of a Hermitian matrix A given the trace of all its powers? [closed]
According to Specht's theorem, a necessary and sufficient condition for unitary equivalent of two matrices $A$ and $B$ is that $\text{Tr} W(A,A^{\dagger})=\text{Tr} W(B,B^{\dagger})$ for all words $W$....
- 11
8
votes
1
answer
1k
views
Calculating a curvature tensor by polarization
I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
- 6,871
8
votes
1
answer
811
views
(0,1)-matrix congruence: is it known?
[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...
- 4,229
3
votes
1
answer
540
views
finding positive vector that multiplied by matrix give positive vector
Hi,
I'm looking to solve the following mathematical problem
I have a given matrix $A$. i want to know if there is a vector $x$ that satisfy two conditions:
the coordinates of $x$ are positive.
the ...
CommunityBot
- 1
11
votes
0
answers
632
views
An elementary linear algebra problem
Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq V_0$...
- 4,015
3
votes
1
answer
829
views
polynomial matrices and its spectrum
Hello, all!
I have a polynomial non-singular square matrix over $\mathbf{F} _q[x]$,
$$\underset{l \times l}{G(x)} = \left( \begin{matrix} g _{0,0}(x) & g _{0,1}(x) & \ldots & g _{0,l-1}(...
CommunityBot
- 1
14
votes
2
answers
2k
views
Finding minimum (or maximum) element of a low rank matrix.
Let $A\in\mathbb{R}^{n\times n}$ and suppose that $A$ is of rank $m\leq n$. Moreover suppose we know $u_1,\ldots, u_m \in\mathbb{R}^{n\times 1}$ and $v_1,\ldots, v_m \in\mathbb{R}^{n\times 1}$ such ...
- 410
3
votes
2
answers
4k
views
Sums of Unitary Matrices
Let $J$ be the $n$ by $n$ matrix whose each entry is $1$. Also define $f(n)$ to be the least $m$ so that there is a $\lambda>0$ so that $\lambda J$ is the sum of at most $m$ unitary matrices. Note $...
CommunityBot
- 1
2
votes
1
answer
2k
views
How to do (m)Gram-Schmidt orthogonalization with integers ? (real life problem) ("mathematicalized reformulation")
New edition of the question, "mathematicalized" (thanks to Gerhard).
Consider and integer valued n*n matrix M, with integers elements in the range -N < m < N.
I want to find integer-valued ...
CommunityBot
- 1
3
votes
0
answers
195
views
Vector spaces over a field of prime order with certain hyperplanes
Let $V$ be a vector space of finite dimentional $d$ over a field of prime order $p$.
For what values of $d$ and $p$, one can find $d+1$ (pairwise distinct) hyperplanes (subspaces of dimension $d-1$) $...
- 3,738
1
vote
1
answer
3k
views
Convergence of Eigenvalues
Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest ...
CommunityBot
- 1
2
votes
3
answers
1k
views
Invariant complement to invariant subspace.
Let $G$ be a compact group and $\rho: G \to End(U)$ its linear representation in a finite dimensional vector space $U$. Fix $V \subset U$ - a subspace invariant under $\rho(G)$. Then it is well known ...
- 405
8
votes
3
answers
1k
views
Are nilpotent orbits degenerations of semi-simple orbits ?
"Examples first:"
Consider so(3,C). (Co)Adjoint Orbits can be described by equations
x^2+y^2+z^2 = R.
R=0 - is nilpotent cone - algebraic closure of the orbit of nilpotent element. (It is union of ...
CommunityBot
- 1
4
votes
0
answers
189
views
Slices of Simplices that are Simplices, Reference?
I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...
- 41
1
vote
1
answer
370
views
A positive semidefinite programming problem
Dear all,
I've got a SDP problem as follows:
$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - {\bf a}^{\top}{\bf H}{\bf b}$,
where ${\bf a}$ and ${\bf b}$ are two constant vectors. May somebody tell ...
- 145
7
votes
1
answer
539
views
A Linear Algebra Problem
Given a matrix $A\in \mathbb{R}^{n\times n}$, I am looking for
a symmetric matrix $S\in\mathbb{R}^{n\times n}$ such that
$$
S A + A^T S = I
$$
$A$ can be assumed to be regular (with positive ...
- 979
2
votes
1
answer
442
views
FFT based algorithm for special matrices
Contest problems with connections to deeper mathematics
This question is with regard to Elkies' answer to the above post.
Vandermonde determinant can be computed using FFT techniques.
Can Moore ...
CommunityBot
- 1
2
votes
2
answers
261
views
Naive tomography question
Given a vectorspace $V$ (over a field $F$) with a specified basis $b_1, \ldots b_n$ and a set $S \subset V$ with two properties:
1) $S$ is a union of lines through the origin (so for all $s \in S$ we ...
CommunityBot
- 1
8
votes
3
answers
510
views
Equitable Allocation of Individuals to Positions
I'm not a mathematician but I working on a problem that feels like it an example of a more general kind of problem and I'm hoping that someone might be able to point me in the right direction.
The ...
- 1,125
1
vote
0
answers
538
views
Representing vertices of a cube using linear combination of tensor product of smaller cubes
Let $n,N \in \mathbb{N}$ with $N \ge n^{2}$.
Let $F[i] = \square[i]$ refer to the cube which has vertices from $\{-1,0,1\}^{n^{i}}$ ($n^{i}$ tuple of alphabets from $\{-1,0,1\} = \square[0] = F[0]$)
...
- 800
2
votes
1
answer
1k
views
On an eigenvalue inequality
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
CommunityBot
- 1
1
vote
1
answer
2k
views
Can one efficiently optimize over the inverse of matrix?
Hello,
I have the following problem:
Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), ...
- 54.2k
0
votes
1
answer
655
views
Fuzzy vector similarity
Hi all,
I have two multi-dimensional vectors representing documents $\vec{a}$ and $\vec{b}$.
Considering cases where there is no overlap between $a$ and $b$ ($a \cap b = \emptyset $), traditional ...
- 16
2
votes
1
answer
816
views
The generator polynomial of cyclic code
Let $q$ be a power of prime number $p$ and let $F_{q^2}$ be a finite field of order $q^2$.
Suppose that "-" be a conjugation operation that is defined as follow:
$-:F_{q^2} \longrightarrow F_{q^2}$...
- 4,784
3
votes
0
answers
919
views
Linear independence over Q of logarithmic powers of prime numbers
I denote $p_k$ the $k^{th}$ prime number ($p_1=2$, etc...)
Clearly, for any $n\in \mathbb{N}^*$, $(\log p_k)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$.
My question concerns a ...
- 101
8
votes
1
answer
5k
views
Constructing a unitary matrix
Setting:
Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: $Q^{...
CommunityBot
- 1
3
votes
0
answers
107
views
Linear relations with small coefficients
NOTE: Slightly more general question follows my specific one at the top
For $1 \leq i,j \leq n$, let $e_{ij}$ be the vector in $\mathbb{R}^{n^{2}}$ with a 1 in entry $(n-1)i + j$, and 0's everywhere ...
- 83
1
vote
2
answers
2k
views
Periodic matrices in SL(3,Z)
Periodic matrices in SL(3,Z) will be conjugated to
product of periodic matrices in SL(2,Z) by +- indentity on a third
integer direction. Is this true?
Sorry, following your comments, maybe ...
- 55.9k
3
votes
0
answers
527
views
3-SAT and a matrix of linear forms representing a non-degenerate matrix
This is a follow-up to the previous question on the same topic. Thanks to Emil Jeřábek I can now ask a more specific question.
As before, let $k$ be a field with $p$ elements. Consider the ...
CommunityBot
- 1
4
votes
1
answer
842
views
determining if a matrix of linear forms represents a non-degenerate matrix
Let $k$ be a field with $p$ elements. Consider the following computational problem
Input: a natural number $n$, $n^2$ linear forms $M_{ij}$, $i,j=1,\ldots n$ in $n^2$ variables $X_{11}, \ldots X_{...
CommunityBot
- 1
2
votes
2
answers
599
views
Eigenvectors of a diagonalizable matrix
Suppose we have a n-by-n symmetric matrix K which can be factorized in a way, K = H * L * H', where L is a m-by-m diagonal matrix and H is a n-by-m matrix. In addition, let's assume n <= m.
Can we ...
CommunityBot
- 1
2
votes
1
answer
453
views
eigen-decomposition of a special companion matrix
I have a special type of companion matrix, where the "special" part is that each element in the matrix are matrices. For instance, the diagonal with "1":s is instead a diagonal with identity matrices, ...
- 23
0
votes
1
answer
409
views
Need help to find an efficient algorithm for the following problem!
Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...
- 375
1
vote
1
answer
1k
views
Characterizing the set of self-orthogonal complex vectors
Let $v\in \mathbb{C}^n$ be an $n$ dimensional complex vector. Define the non-standard bilinear form $\left< u,v \right> = u^T v$ (the usual inner product except without the conjugation). What ...
- 6,614
5
votes
1
answer
637
views
"Orthogonal complement" in $\mathbb{Z}_q^n$
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the integers mod $q$. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w \...
- 56.6k
3
votes
3
answers
3k
views
Generalization of eigenvalues/vectors to modules?
What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...
- 9,132
3
votes
0
answers
390
views
Approximate action of unitary matrix with permutation matrix
Given a unitary matrix Q and a symmetric matrix B, I am trying to find a permutation matrix P such that
$ || QBQ^{T} - PBP^{T} ||_{F} $
is minimized.
The straightforward method of minimizing $ ...
