All Questions
6,289 questions
25
votes
16
answers
4k
views
functions satisfying "one-one iff onto"
Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...
7
votes
2
answers
3k
views
Factorizing a block symmetric matrix
Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible.
I would like to ...
- 20.2k
4
votes
1
answer
314
views
Spectral Properties of $A(I-A)^{-1}$
I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
- 8,512
6
votes
4
answers
1k
views
Polar decomposition for quaternionic matrices?
A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...
- 3,979
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 $f^n(m)=0$. ...
- 63.1k
2
votes
0
answers
769
views
minimum rank of a general matrix
I have a general $m\times m$ matrix whose antisymmetric part is non singular. What is the minimum rank of that matrix?
- 21
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 ...
- 185
3
votes
2
answers
2k
views
Trace Determinant [closed]
Let $A$ be an $n \times n$ matrix. Are there formulas that convert linear combinations of traces of powers of $A$ to determinant of $A$ and vice versa from linear combinations of determinants of ...
- 13.9k
8
votes
2
answers
583
views
Efficiently computing a few localized eigenvectors
Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
- 817
6
votes
3
answers
2k
views
Lipschitz continuity of singular values
How smooth are the singular values of a matrix $F$ in terms of entries of $F$? I am hoping for Lipschitz continuity, but was not able to find it.
- 63
28
votes
4
answers
5k
views
Jacobi's equality between complementary minors of inverse matrices
What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...
- 4,466
4
votes
1
answer
977
views
Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
- 41
6
votes
2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 1,649
1
vote
1
answer
720
views
A question on gauge functions
In the second paragraph on Page 71 of the book Matrix Analysis by
Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 from
...
- 13
0
votes
3
answers
7k
views
Find an $N$-dimensional vector orthogonal to a given vector
I'm writing an eigensolver and I'm trying to generate a guess for the next iteration in the solve that is orthogonal to to all known eigenvectors calculated thus far. This means that if I have only ...
7
votes
1
answer
530
views
Do real vectors attain matrix norms?
I apologize if the following question ends up being too elementary for this website; I asked it on math.SE a week ago and it remains unanswered.
Let $A$ be an $n \times n$ matrix with real entries ...
- 571
3
votes
2
answers
2k
views
When can an eigenvector be chosen uniquely which is invariant to permutation?
Suppose $A\in\Re^{n\times n}_{sym}$ is a symmetric matrix with eigenvalues $\lambda_1,\dotsc,\lambda_n$ in decreasing order. What I seek is a way to choose an eigenvector that is invariant to ...
- 53
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
- 1,222
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 ...
- 637
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 \...
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(\...
- 2,274
3
votes
3
answers
310
views
measuring $n\ 2$-planes in $\mathbb{R}^{2n}$
Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ ...
- 2,274
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
1
vote
2
answers
307
views
Continuous family of Markov chains
Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition ...
- 1,069
1
vote
2
answers
748
views
Existence of polynomial equation system solution
For $1 \leq i \leq n$, let
$A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nn} \\
\end{bmatrix}$, $B_i=\begin{bmatrix} b_{i1} \...
- 57
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 ...
- 31
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
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
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
- 181
5
votes
1
answer
603
views
Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups
I am reading the Kleidman–Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost ...
- 731
1
vote
1
answer
869
views
Shortcut for variance of vectors
I have a dataset of vectors and need to find the sum of squares or variance based on the euclidean distance between the vectors.
I can do this by finding the "average" vector (by calculating the ...
- 21
2
votes
2
answers
820
views
Computing the multiplicity of an eigenvalue of a 0-1 symmetric matrix...
When we want to compute the multiplicity of an eigenvalue of a 0-1 symmetric matrix (viewed as the adjacency matrix of an undirected regular graph), we commonly resort to the know lemma of Feit and ...
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 $...
- 83
12
votes
5
answers
2k
views
analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 19.7k
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
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 ...
- 329
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 ...
- 24.9k
12
votes
1
answer
1k
views
Decomposition of positive definite matrices.
It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...
12
votes
2
answers
1k
views
Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 123
4
votes
0
answers
189
views
Slices of Simplices that are Simplices, Reference?
I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...
- 41
12
votes
3
answers
4k
views
Status of Hadamard matrix conjecture
I would like to know if any progress has been made on Hadamard conjecture :
Hadamard matrix of order $4k$ exists for every positive integer $k$.
- 121
7
votes
1
answer
539
views
A Linear Algebra Problem
Given a matrix $A\in \mathbb{R}^{n\times n}$, I am looking for
a symmetric matrix $S\in\mathbb{R}^{n\times n}$ such that
$$
S A + A^T S = I
$$
$A$ can be assumed to be regular (with positive ...
- 979
1
vote
1
answer
370
views
A positive semidefinite programming problem
Dear all,
I've got a SDP problem as follows:
$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - {\bf a}^{\top}{\bf H}{\bf b}$,
where ${\bf a}$ and ${\bf b}$ are two constant vectors. May somebody tell ...
- 145
2
votes
1
answer
313
views
Derivation of Iteration Rules
Suppose we are given a matrix $V$ and our goal is to find non-negative matrices $W$ and $H$ such that $V \approx WH$. So we want to minimize $K(V || WH)$ (Kullback-Leibler Divergence) where $$K(V||WH) ...
- 21
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 ...
- 201
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 ...
- 317
10
votes
1
answer
923
views
Conjugation between commutative subalgebras of a matrix algebra?
Let $K$ be an algebraically closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two isomorphic commutative subalgebras of $M_n(K)$, it is true that there ...
- 545
8
votes
0
answers
694
views
Path connected set of matrices?
Consider the collection of $n$ by $n$ matrices
$$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$
where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint ...
- 1,533