All Questions
6,177 questions
0
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Positivity and symmetrization
Let $A$ be a symmetric positive matrix, and let $B$ be invertible. Is
$$BAB^{-1} + B^{-1}AB$$
always positive?
Let $C$ be a real matrix with real positive spectrum. Is
$$C + C^T$$
positive?
Are ...
- 28.6k
0
votes
1
answer
130
views
Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 39.9k
0
votes
0
answers
429
views
[]-infinity algebra and Projective representation
This is a very vague question.
We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...
- 1,271
27
votes
1
answer
4k
views
If $V$ is a vector space with a basis. $W\subseteq V$ has to have a basis too?
Suppose $V$ is a vector space, we say that $\mathcal B$ is a basis for $V$ if:
Every $v\in V$ can be written as a linear combination of elements of $\mathcal B$;
If $\sum\alpha_i b_i = 0$, where $\...
- 14k
9
votes
5
answers
2k
views
Rank of a free module without the axiom of choice
Perhaps my question is really naive. I teach this semester in algebra. I am embarassed about the proof that a free module over an integral domain has a well-defined rank. It is based on the theorem ...
CommunityBot
- 1
15
votes
3
answers
1k
views
The normalizer of $\mathrm{GL}(n,\mathbf Z)$ in $\mathrm{GL}(n,\mathbf Q)$
It seems that the normalizer of $H=\mathrm{GL}(n,\mathbf Z)$ in $G=\mathrm{GL}(n,\mathbf Q)$ is "almost" equal to itself, that is,
$$
N_G(\mathrm{GL}(n,\mathbf Z))=Z(G) \cdot \mathrm{GL}(n,\mathbf Z)
...
CommunityBot
- 1
3
votes
2
answers
5k
views
Volume change under linear transformation
It is well-known, that given a linear transformation $f \colon \mathbb R^n \rightarrow \mathbb R^m$, where $m \ge n$, the $m$-dimensional volume of an image of any measurable subset $S \subseteq \...
- 32.4k
1
vote
2
answers
313
views
How many DISTINCT vectors we get from pairs v_i + v_j for some set of given vectors v_i ?
Consider some set of vectors v_i i=1...N , v_i \in Z^k.
e.g. N = 10^4; k = 10
Consider all possible sums: v_i + v_j.
Is it possible to estimate how many DISTINCT vectors we get in advance without ...
- 39.9k
10
votes
1
answer
1k
views
Intrinsic description of the image of $V \to V^{**}$
Let $V$ be a vector space over a field $K$. Call a linear map $F : V^* \to K$ representable if there is some $v \in V$ such that $F(w) = \langle w,v \rangle$ for all $w \in V^*$. Here, $\langle w,v \...
- 43.2k
1
vote
3
answers
640
views
Eigenvalues of Krylov matrices
Let an $n\times n$ matrix ${\bf A}$, the all ones vector ${\bf w}$, and the $n\times n$ Krylov matrix
$${\bf K}_n = \left[ {\bf w}\;\;{\bf A}{\bf w}\;\;\ldots \;\; {\bf A}^{n-1}{\bf w}\right].$$
Is ...
- 1,171
20
votes
3
answers
6k
views
When is $\ker AB = \ker A + \ker B$?
Prove/ Disprove: Let $n$ be a positive integer. Let $A$, $B$ be two $n \times n$ square matrices over the complex numbers. If $AB = BA$ and $\ker A = \ker A^2$ and $\ker B = \ker B^2$
then $\ker AB = ...
- 685
10
votes
1
answer
607
views
Properties of a matrix-valued generalization of the $\Gamma$ function
I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$....
- 28.6k
6
votes
5
answers
2k
views
Computer algebra system for calculation of characteristic polynomial of sparse matrix
I have a $n \times n$ matrix, for which i need to calculate the characteristic polynomial. The matrix is over $GF(2)$, and $n \approx 10^4$. However the matrix is very sparse, with around $ n $ non ...
- 133
5
votes
1
answer
419
views
positive hermitian elements in $M_n(\mathbb{C})$
Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers
have some special properties:
(i) they are closed under sum,
(ii) they are closed under multiplication by positive ...
- 1,887
1
vote
1
answer
2k
views
Find vector in R^n which is orthogonal to given (n-1) vectors v_i under condition v_i are orthonormal.
If we need to find vector in R^n which is orthogonal to given (n-1) vectors,
this is basically solving linear system of equations and can be done in O(n^3) operation.
I wonder is there some ...
- 61.9k
1
vote
0
answers
197
views
Matrix Theory approach to general Linear Equations over skew fields
Is there a matrix way of writing system of linear equation over a skew field where the variables in the equations are both left multiplied and right multiplied by elements of the skew fields.
Is the ...
- 113
9
votes
1
answer
352
views
A Family of Bases for a Vector Space
Let $V$ be a multiset of $kn$ nonzero vectors in $\mathbf{R}^n$. Suppose that for $1 \leq d \leq n$, each $d$-dimensional subspace of $\mathbf{R}^n$ contains at most $kd$ members of $V$. Then $V$ ...
- 56.6k
3
votes
2
answers
194
views
Cases of almost-linear time solvable linear systems
Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$
Solving for ${\bf x}$ through standard Gaussian Elimination ...
- 20.2k
0
votes
1
answer
2k
views
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 ...
CommunityBot
- 1
6
votes
3
answers
2k
views
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 $...
- 156k
3
votes
4
answers
1k
views
How can I generate (suitably random) symplectic matrices?
I would like to write a computer script to generate a lot of symplectic matrices. How can I do this? Is there a parameterization of all symplectic matrices?
- 4,466
1
vote
0
answers
128
views
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}...
CommunityBot
- 1
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
2
votes
1
answer
2k
views
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 ...
- 16.2k
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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, ...
- 37.4k
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 ...
- 28.6k
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}}$ ...
- 52.3k
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 ...
- 37.7k
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 ...
- 188k
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. ...
CommunityBot
- 1
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) \\
\...
- 96.4k
10
votes
2
answers
3k
views
Maximizing the Smallest Eigenvalue of a Diagonally Dominant Matrix
Assume that we have a full-rank diagonally dominant matrix $A$, all the diagonal elements of which are positive, all the non-diagonal elements are negative, and the sum of the absolute values of the ...
CommunityBot
- 1
5
votes
2
answers
612
views
A Boolean function that is not constant on affine subspaces of large enough dimension
I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of ...
CommunityBot
- 1
4
votes
2
answers
323
views
Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?
Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that
$$\lim_{n} ~a_{nn} = L_1, \qquad
\lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$
exist. Is there a sufficient criteria to conclude ...
- 4,729
5
votes
2
answers
487
views
Some weird "system" of inequalities in nonnegative integers.
Suppose I have a bunch of nonnegative integers $(a_{ijkl})_{1 \leq i \leq j \leq k \leq l \leq 17}$ such that for all 17-tuples nonnegative integers $w_t$ (for $1 \leq t \leq 17$) we have that $$\min_{...
- 34.3k
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 ...
- 5,743
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 $A$, $B$ ...
- 96.4k
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 ...
- 263
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:...
- 32.4k
8
votes
4
answers
3k
views
Finite dimensional vector spaces over a complete but not-necessarily-valued field
I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
CommunityBot
- 1
3
votes
2
answers
753
views
The Action of the inverse of a shifted matrix on a vector
This question concerns a matrix $A\in\mathbb{C}^{n\times n}$, whose inverse we know the action of on a particular vector $v\in\mathbb{C}^{n}$. If we know that $A^{-1}v = u$, is there any way to ...
CommunityBot
- 1
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 ...
- 12.1k
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
2
votes
1
answer
821
views
Question about orthogonal matching pursuit
Let y be a n-vector, X a n-by-p matrix of full rank (p < n) and b a p-vector, so that y = Xb + e, for some noise vector e. I am not sure how to show reduction of error in orthogonal matching ...
CommunityBot
- 1
5
votes
2
answers
475
views
Can finitely many hermitian (positive-semidefinite) operators always be embedded into a small dimensional space preserving inner product?
Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which ...
CommunityBot
- 1
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