All Questions
6,290 questions
15
votes
3
answers
18k
views
angle between subspaces
Let $E$ be a finite dimensional real inner product space. I want to define the angle between two subspaces $E_1$ and $E_2$. This has a fairly obvious meaning if $E_1$ is 1-diemsnional: Take the ...
- 421
3
votes
4
answers
570
views
A polynomial homomorphism from Gl to the group of units is a power of the determinant
I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism $\mathrm{Gl}...
- 4,902
1
vote
0
answers
158
views
Comparing the volume of a rational lagrangian under a linear symplectomorphism.
Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace $...
4
votes
2
answers
496
views
Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?
I need to answer (affirmatively, I hope) the following question:
In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset
...
- 1,804
6
votes
1
answer
363
views
Grouping vectors together
Given $n$ unit vectors in $\mathbb{R}^n$ s.t. $0 \leq u\cdot v<1$ for all pair of distinct vectors $u,v$. These vectors span a $d$-dimensional subspace s.t. $d< n$. We conjecture that it is ...
- 505
4
votes
0
answers
190
views
Large scale sparse system of linear equations
What is the best know algorithm for solving a large sparse system of linear equations? The system I'm working on is not symmetric, not positive definite and integer. The only benefit is being sparse. ...
- 221
21
votes
4
answers
3k
views
Computing the Zariski closure of a subgroup of SL(n,Z)
Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...
- 3,201
3
votes
0
answers
156
views
inverse M-matrix times mixed-sign vector
Recently a colleague and I came across this unusual phenomenon.
Take $M\in\mathbb{R}^{n\times n}$ a singular irreducible M-matrix, and $b\in\mathbb{R}^{n}$ such that the system $Mx=b$ is solvable (so,...
- 20.2k
1
vote
2
answers
495
views
singular value decomposition
Given regular matrices $A_i,B_i \in \textrm{GL}_n(\mathbb{R}),$ $i=1,2$.
Let $A_1 = U_1 B_1 V_1$ and $A_2=U_2 B_2 V_2$ where $U_i,V_i \in \textrm{O}_n(\mathbb{R})$ $(i=1,2)$ are orthogonal matrices. ...
- 175
5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 922
1
vote
0
answers
413
views
Combinatorial Interpretation of an Extension of Gaussian Polynomials
It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over $...
- 443
11
votes
2
answers
2k
views
Groups of matrices that preserve several quadratic forms
Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q_i=U Q_i U^T$, $i=1,2..,k$ What is known about such groups? (at least ...
- 2,232
1
vote
2
answers
332
views
determinantal identity sought
Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I_{n}+AB)=det(I_{m}+BA)$.
Is there an analogous identity of the form $det(P_{1}+AB)=det(P_{2}+BA)$, ...
- 6,962
4
votes
1
answer
471
views
Ask for theory about the weighted L^2(R^d) space.
Dear MOs,
I am now considering the following norm:
$$
||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.
$$
where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
- 1,649
2
votes
0
answers
372
views
What is the Birkhoff norm of a Perron vector?
Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...
- 6,962
1
vote
0
answers
218
views
Singular quadratic space
Let $(V,b)$ a symmetric bilinear space. An old theorem of Witt says that if $(V,b)$ is regular, then given a subspace $W$ of $V$ and an isometry $\sigma: W \to V$, there exists an isometry $\Sigma: V \...
- 5,163
2
votes
2
answers
598
views
On Random Vectors and Eigenvectors of Symmetric Matrices
I have a question that might be answered with a pointer to some references or with some discussion. I did some searching, to no avail, but I realized that I might not have the vocabulary to form a ...
- 325
8
votes
1
answer
490
views
ad (A^n) is a polynomial in ad A ?
Let $k$ be a field and $n$ a nonnegative integer. For any matrix $U\in\mathrm{M}_n\left(k\right)$, let $\mathrm{ad} U$ denote the map $\mathrm{M}_n\left(k\right)\to \mathrm{M}_n\left(k\right),\ V\...
- 33.8k
5
votes
1
answer
476
views
Elementary Markov Chain Question
Are any general conditions known on a finite transition nxn matrix that ensure that there exists at least one mth root which is also a transition matrix? It is easy to construct a 3x3 , diagonally ...
- 313
4
votes
2
answers
3k
views
Number of Jordan canonical forms for an nxn matrix
How many Jordan canonical forms may have an nxn matrix?
In the article https://oeis.org/A000219 states that the number of Jordan canonical forms for an nxn matrix is the Number of planar partitions ...
- 95
6
votes
1
answer
463
views
Is the solution of this linear system always positive definite?
Let $P\in \mathbb{R}^{n\times r}$ be a submatrix (which consists of the first $r$ columns) of an arbitrary $n\times n$ orthogonal matrix ($1 < r < n$). Let $I_n$ denote the $n\times n$ identity ...
- 227
2
votes
0
answers
230
views
Consistency of a system of linear equations
I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...
- 221
5
votes
2
answers
315
views
Bounding the minimum entry of an inverse matrix
Suppose $A$ is an $n\times n$ stochastic matrix, that is, entrywise nonnegative and row sums are all $1$. If $A$ is invertible, is it true that the minimum diagonal entry of $A^{-1}$ is no larger than ...
- 478
1
vote
2
answers
3k
views
Null space vs. semi-positive definite matrix
Defining the right generalized inverse of a non-square Jacobian matrix $J$, $J^{\#}$, as
$J^{\#} = M^{-1} J^T \left(J M^{-1} J^T\right)^{-1}$
where the matrix $M \succ 0$ is positive definite and ...
- 13
1
vote
0
answers
629
views
Totally unimodular Matrices
A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
- 11
3
votes
2
answers
673
views
Asymptotic number of invertible matrices with integer entries
Let $\|\cdot \|$ be some matrix norm on the space of $n \times n$ matrices. Denote
$$ M(r) := \{ A \in \mathrm{Mat}_{n \times n}(\mathbb{Z}) \mid \| M \| \leq r \}.$$
Denote by $p(r)$ the fraction of ...
- 9,853
5
votes
4
answers
1k
views
Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before?
I'm investigating the eigenvalue ratios
$$
\frac{\lambda_1}{\sum_{j=2}^N\lambda_j}
\quad\mbox{and}\quad
\frac{\sum_{j=1}^N\lambda_j}{\sum_{j=2}^N\lambda_j}
$$
of the NxN matrix $B=AA^T$. $\lambda_1$ ...
- 161
2
votes
1
answer
8k
views
Properties of eigenvalues of general nonnegative matrices
I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the ...
- 83
2
votes
0
answers
291
views
Eigen-decomposition perturbation
Let $A$, $B$ and $A_k + B$ be symmetric matrices with eigenvalues $\sigma_1 \geq \sigma_2 \ldots \geq \sigma_n$, $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ and $\lambda_1 \geq \lambda_2 \ldots \geq \...
- 21
1
vote
1
answer
178
views
inequality for a symmetric nonnegative matrix
Given $A$ symmetric and semidefinite positive, for each $x$
$$ x'Ax \geq \frac{1}{\Vert A\Vert} \Vert Ax \Vert^2 $$
This inequality appears at page 24 of "Introduction to Optimization" from Boris T. ...
- 63
3
votes
2
answers
1k
views
Is there a different construction of "the" tensor product of two modules?
It may be a pseudo question. But I still decide to ask. Given two $k$-modules $M$ and $N$，it seems to me that in the literature the tensor product $M\bigotimes_kN$ is always defined as the quotient of ...
3
votes
2
answers
195
views
Inflate a simplex, change rows to make the rank n
I have a simplex, n + 1 points in $\mathbb{R}^n$,
which may have rank $r < n$.
Is there a cheap way of "inflating" it to rank $n$,
changing a few, all but $r$, of the points ?
The points are ...
- 265
4
votes
2
answers
2k
views
Woodbury formula
I wonder - do you know of any example where the Woodbury formula (cf. http://en.wikipedia.org/wiki/Woodbury_matrix_identity) was crucially used to prove anything?
It might be a useful computational ...
- 6,962
11
votes
2
answers
9k
views
How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?
I have a matrix $K$ which is the sum of a identity and a Kronecker product of two symmetric matrices as following and I want to calculate the inverse of it $K^{-1}$.
\begin{eqnarray}
K=\mathbf{I}_{mn}+...
- 607
6
votes
1
answer
333
views
construction of matrices verifying an identity
Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that B is nonsingular and $AB\neq BA$. Can we always find real numbers $t_1,⋯,t_p$ such that
$$B\left(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\right)A=A\...
0
votes
1
answer
353
views
Moore-Penrose bound question
Suppose that we are given an equation $Ax=b$. The minimum least-squares solution is of course $x_{m}=A^{\dagger}b$. What I want to know is whether there are known bounds on $||x-x_{m}||$. In the ...
- 6,962
1
vote
0
answers
443
views
Diagonalizing matrix with a special conjugate transpose property
Hi all,
I'm looking for the minimum criterion on $A\in M_{3x3}(\mathbb{C})$ (a $3x3$ complex matrix) such that:
1) $A$ is diagonalizable by a matrix $T\in M_{3x3}(\mathbb{C})$
2) $T$ is such that $...
- 111
1
vote
2
answers
491
views
Simultaneous Smith Normalization of a Composable Matrix Sequence
Let $\mathsf{R}$ be a PID and consider a collection of free, finitely generated $\mathsf{R}$-modules $V_1,\ldots,V_n$ along with module maps $m_j:V_j \to V_{j+1}$. That is, we have the following ...
- 15.5k
5
votes
2
answers
712
views
Maximum size of $k$-wise linearly independent set within $\lbrace 1, 2, 3, ..., u \rbrace^k$
Given a positive integer $u$, how many $k$-dimensional vectors whose coordinates are all in $\lbrace 1, 2, 3, ..., u\rbrace$ can you choose so that any $k$ of them are linearly independent? ...
- 53
1
vote
1
answer
157
views
Augmenting sub-spaces through a basis
Let $t \lt n-1$,
A family { $V_1, V_2, ..., V_n$ } sub-spaces of an $n$-dimensional vector space $V$ is called $t$-feasible if it satisfies conditions (i) and (ii) below:
(i) $\dim(V_i) = t$, for ...
- 1,034
5
votes
4
answers
1k
views
determinants and polynomials in matrices
Muirhead (1982, "Aspects of Multivariate Statistical Theory") references on page 59
a result (from MacDuffee, 1943, chap 3, "Vectors and Matrices") a book I cannot find):
" The only polynomials in ...
- 2,600
6
votes
2
answers
744
views
Unicity of a vector space frame's dual frame
The Wikipedia page on vector space frames gives a construction to find a dual frame for a given frame. Specifically, given a set of vectors $\{ e_k \}$ in a Hilbert space $\mathcal{H}$ such that for ...
- 2,358
0
votes
0
answers
161
views
vector equation
Suppose you have an equation of the form $Hx=Ky$, where $x,y$ are vectors of length $n,m$ respectively ($m>n$) and $H,K$ are matrices of orders $n \times n,n \times m$ respectively. Is there some ...
- 6,962
6
votes
1
answer
453
views
Computing the correlation between two vectors without divulging them
Alice and Bob respectively know a vector of $N$ real numbers $u$ and $v$. They would both like to know $\rho = \langle u,v \rangle/N$ but Alice does not want Bob to gain anymore information about $u$ ...
- 1,902
16
votes
1
answer
2k
views
Moore-Penrose Inverse as an adjoint
A Moore-Penrose pseudoinverse of a morphism $f: V \rightarrow W$ between Euclidean vector spaces is a map $g: W \rightarrow V$ in the other direction satisfying the identities
$fgf = f$
$gfg = g$
$(...
- 1,704
10
votes
2
answers
4k
views
Perturbation theory for the generalized eigenvalue problem
Is there a standard reference for the perturbation theory of the generalized eigenvalue problem?
More specifically, I would like to get a systematic expansion for the problem
$(A_0 + \epsilon A_1)...
- 1,193
10
votes
5
answers
3k
views
Generalization of the polarisation formula for symmetric bilinear forms to symmetric multilinear forms
Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vector space and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$.
The Polarisation Formula ...
- 109
31
votes
4
answers
2k
views
Probability of zero in a random matrix
Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, ...
- 37.7k
9
votes
3
answers
486
views
Representing a real number as the value of a countably infinite game
Is it true that for any real number $p$ between 0 and 1, there exist finite or infinite sequences $x_m$ and $y_n$ of positive real numbers, and a finite or infinite matrix of numbers $\varphi_{mn}$ ...
- 187
6
votes
2
answers
1k
views
Simple Lie algebras and Jordan decomposition
Let $F$ be an algebraically closed field and let $L$ be a simple Lie algebra of dimension $n$ over $F$. Let $ad: L\longrightarrow End_F(L)$ denote the adjoint representation of $L$. If $F$ has ...
- 630