All Questions
6,289 questions
1
vote
1
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Find vector in R^n which is orthogonal to given (n-1) vectors v_i under condition v_i are orthonormal.
If we need to find vector in R^n which is orthogonal to given (n-1) vectors,
this is basically solving linear system of equations and can be done in O(n^3) operation.
I wonder is there some ...
- 24.9k
1
vote
0
answers
197
views
Matrix Theory approach to general Linear Equations over skew fields
Is there a matrix way of writing system of linear equation over a skew field where the variables in the equations are both left multiplied and right multiplied by elements of the skew fields.
Is the ...
- 113
6
votes
5
answers
2k
views
Computer algebra system for calculation of characteristic polynomial of sparse matrix
I have a $n \times n$ matrix, for which i need to calculate the characteristic polynomial. The matrix is over $GF(2)$, and $n \approx 10^4$. However the matrix is very sparse, with around $ n $ non ...
- 61
3
votes
2
answers
195
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Cases of almost-linear time solvable linear systems
Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$
Solving for ${\bf x}$ through standard Gaussian Elimination ...
- 449
10
votes
1
answer
607
views
Properties of a matrix-valued generalization of the $\Gamma$ function
I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$....
- 1,243
4
votes
1
answer
492
views
Generalization of the "double cap conjecture" to a vector space with complex field
The conjecture that I proposed in
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun.
See for ...
- 1,207
8
votes
0
answers
544
views
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
Let R be a region on a hypersphere. Each point A of the hypersphere
is associated with a vector pointing to A and with origin at
the centre of the hypersphere. So let me identify each point with a
...
- 1,207
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^{...
- 605
30
votes
2
answers
2k
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When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
- 418
0
votes
1
answer
2k
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True divide and conquer inversion of large matrices
In https://math.stackexchange.com/questions/2735/solving-very-large-matrices-in-pieces there is a way shown to solve matrix inversion in pieces. Is it possible to turn it into a true divide and ...
Countably Infinite
10
votes
2
answers
7k
views
Bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?
$\DeclareMathOperator\Tr{Tr}$Let $A_i$ with $i=1,\dotsc,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $\Tr(p)=1$. Let $0< a_i<1$ and $\...
- 103
6
votes
3
answers
2k
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Finding the action of the symplectic group on the Siegel-half plane
Let $S$ be the Siegel-half plane of dimension $n$, i.e. the set of complex $n \times n$ matrices $Z$ which are symmetric and whose imaginary part is positive-definite. In dimension 1 we can identify $...
- 4,343
1
vote
0
answers
128
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Spectral decomposition of R matrix -> Wenzl projectors?
Just curious: if you take a R matrix from knot theory and apply
a spectral decomposition (see. e.g. my following post
Matrix decomposition the other way)
you'll get projectors: $T_i*T_j=T_i*\delta_{ij}...
- 4,829
16
votes
2
answers
9k
views
Solving a quadratic matrix equation
This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? ...
- 1,538
3
votes
0
answers
212
views
Unique structures in a class of connected directed hypergraphs
Edit: The problem has been slightly revised, as I discovered that one of the questions I asked has an answer in the negative.
I'm working in a setting involving constraints on a system described by a ...
- 1,444
0
votes
0
answers
157
views
Matrices satisfying certain pair-wise constraints
Consider given pairs of variables: $a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints:
$\sum_{r=1}^...
- 13.9k
14
votes
1
answer
4k
views
Do these matrix rings have non-zero elements that are neither units nor zero divisors?
First, a disclaimer: This is a repost of a question I asked on stackexchange (no answer there).
Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with ...
- 1,197
25
votes
5
answers
2k
views
When is a matrix power nonnegative
The following question came up today during a discussion:
Suppose $A$ is an $n \times n$ real matrix. Is there some way to tell whether there exists an integer $q > 0$ such that $A^q$ is ...
- 28.6k
2
votes
1
answer
2k
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Subgroups of the Euclidean group as semidirect products
Consider the Euclidean group $E(n)$ as the semidirect product for Euclidean vector space $\mathbb{E}^n$ with its orthogonal group $O(\mathbb{E}^n)$:
$E(n)=\mathbb{E}^n\rtimes O(\mathbb{E}^n)$
Then ...
- 347
5
votes
1
answer
196
views
Expressing a element of a Matrix subgroup in terms of subgroup generators
I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...
- 153
1
vote
0
answers
222
views
(Non-)Surjectivity of the Maslov index
Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $...
- 3,605
5
votes
1
answer
641
views
Characterizing invertible nonnegative matrices with bounded sums
Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
- 28.6k
3
votes
0
answers
312
views
Linear complementarity problem: principal pivoting algorithm
I'm trying to implement the "Dantzig; van de Panne and Whinston" principal pivoting algorithm for solving symmetric positive semi-definite LCPs from "The Linear Complementarity Problem" book (...
- 151
3
votes
1
answer
357
views
Mathematical Programming with other Algebras than Linear
Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.
What analogies are there for ...
- 2,383
12
votes
3
answers
1k
views
Conjugacy in $GL(n,\mathbb Z)$
How can I determine whether $A_1,A_2\in GL(n,\mathbb Z)$ conjugate in $GL(n,\mathbb Z)$ and if they are, how can I find a $P\in GL(n,\mathbb Z)$ for which $A_2 = P^{-1}.A_1.P$ ?
In $GL(n,\mathbb Q)$ ...
- 347
5
votes
1
answer
752
views
Convex Analytic or linear algebraic proof that a certain psd matrix is a sum of rank 1 psd matrices
Can you prove the following using techniques from convex analysis or linear algebra? I was originally seeking an elementary proof, but I think it is better to broaden the scope for this bounty ...
user2529
5
votes
2
answers
2k
views
partial Derivatives of Eigen value decomposition or Singular value decomposition
Hi All,
Suppose I've a symmetric matrix $A_{N \times N} = (A_{ij})$ which has a eigen value decomposition $A = UDU'$. I would like to know under what conditions $\frac{\partial U}{\partial A_{ij}}$ ...
- 63
2
votes
2
answers
207
views
Families of quadratic Hamiltonians
Hi. What type of 2n dimensional real symmetric matrices can be diagonalized with symplectic transformations (meaning M->SMS^T, S^T means transpose and S is an element of the 2n dimensional real ...
- 23
2
votes
1
answer
351
views
is there a way to solve the following equation?
(I tried asking that on math.stackexchange.com, but did not get a satisfying answer. I am trying here as well, in case someone here will have more insight. The question was eventually abandoned there. ...
- 21
0
votes
1
answer
346
views
Conditions to the existence of a solution of a system of congruences [closed]
Let $p$ be a prime. Consider the following congruences:
$$
\begin{array}{lcl}
a_1 x & = & c_1 (\text{mod } p) \\\\
\vdots & & \vdots\\\\
a_n x & = & c_n (\text{mod } p) \\
\...
- 141
4
votes
3
answers
755
views
Is this statement about the real edge space of a graph known or trivial?
The statement is:
($u$ is a fixed node in a fixed graph $G$)
$G$ is 3-connected
if and only if
the set of u-cycles span $\mathbb{R}^{E(G)}$.
A u-cycle is a simple (no vertex repetitions) cycle in G ...
- 406
10
votes
4
answers
903
views
Relationship between free probability and deterministic graphs?
Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
\end{...
- 1,890
1
vote
2
answers
1k
views
Measure of Subspace of Matrices with repeated Singular Values
Hi All,
Let us consider a P x Q real matrix (P >= Q). It can be thought of as an element of $\mathbb{R}^{PQ}$. We are considering Lebesgue measure over that space. My question is whether the ...
- 13
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
3
votes
1
answer
624
views
Counting matrices with different determinants
Let $A$ and $B$ be two matrices of order $n$ over a finite subset of integers $S$ such that $A$ and $B$ are positive-definite, nonsingular and symmetric.
I am interested in proprieties about $A$, $B$ ...
- 3,463
3
votes
2
answers
294
views
Does the automorphism group of a cone determine the cone?
A cone is a $R_+$-module. That is, a cone is an abelian monoid that is closed under nonnegative real scalar multiplication. An automorphism of a cone is a bijective $R_+$-linear map. That is a map $f:...
user2529
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 ...
- 3,217
3
votes
1
answer
525
views
Linear and Isometric Automorphism Groups of the PSD Cone
Let $S_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$.
...
user2529
7
votes
1
answer
5k
views
How much can a diagonal matrix change the eigenvalues of a symmetric matrix?
Suppose that we have a symmetric matrix ${\bf S}$ with eigenvalue decomposition ${\bf S} = {\bf Q}{\bf \Lambda}{\bf Q}^T$. Assume that we have two diagonal matrices ${\bf D}_1$ and ${\bf D}_2$ that ...
- 449
9
votes
1
answer
352
views
A Family of Bases for a Vector Space
Let $V$ be a multiset of $kn$ nonzero vectors in $\mathbf{R}^n$. Suppose that for $1 \leq d \leq n$, each $d$-dimensional subspace of $\mathbf{R}^n$ contains at most $kd$ members of $V$. Then $V$ ...
0
votes
1
answer
12k
views
Square matrices: $(A+B)^2=A^2+B^2$ [closed]
If $a, b$ are two numbers such that $(a+b)^2 = a^2 + b^2$, then $a.b = 0$.
Is there a similar statement for square matrices.
"If $A, B$ are square matrices such that $(A+B)^2 = A^2 + B^2$, then $A.B ...
- 2,773
1
vote
0
answers
254
views
Operator eigenvalues and eigenvalues of pointwise evaluation matrix
Let $D\subset\mathbb{R}$ be a bounded interval and $f: D\times D \rightarrow \mathbb{R}$ a real-valued analytic function of two variables such that $f\in L_2(D\times D)$. Suppose we have upper bounds ...
- 3,217
4
votes
0
answers
246
views
Algorithm/denominators of elements of a rational affine space
I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
- 8,524
4
votes
2
answers
818
views
Double orthogonal complement of a finite module
Crossposted from math.stackexchange since I'm not getting any answer.
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...
- 141
1
vote
0
answers
213
views
Eigenvalue distribution of positive-definite analytic function
Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a real-valued symmetric, positive definite function. Let $\{(x_i,y_i)\}_{1\leq i\leq N}\subset[0,1]^2$ be a finite, distinct set of coordinates. The point-...
- 3,217
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 $k$?...
- 5,163
4
votes
2
answers
322
views
Algebra with elements x, y such that r(x)=r(y) for all finite-dimensional modules r
I'm interested in finding an algebra with elements x,y which are identified by every finite-dimensional module. I'm primarily interested in the case that everything is over the complex field, but ...
- 2,513
1
vote
1
answer
459
views
Looking for name of a famous matrix
Let $A_n$ be the $n\times n$ matrix whose $(i,j)$-element is $1/(i+j-1)$. This is a famous matrix in linear algebra and has some nice properties (like, its inverse is integral).
Does anybody remember ...
4
votes
1
answer
298
views
Is there a standard name for the intersection of all maximal linearly independent subsets of a given set in a vector space?
The title more or less says it all.... Let $V$ be a vector space (over your favorite field; $V$ not necessarily finite dimensional), and let $S$ be a subset of $V$. A maximal linearly independent ...
- 12.8k
5
votes
3
answers
543
views
Finding a hyperplane that splits a convex polytope evenly
Say we have a convex polytope in standard form:
\begin{equation*}
\begin{array}{rl}
\mathbf{A}\mathbf{x} = \mathbf{b} \\\\
\mathbf{x} \ge 0
\end{array}
\end{equation*}
Are there any known methods ...
