All Questions
6,292 questions
21
votes
3
answers
51k
views
What is the time complexity of truncated SVD?
Full SVD, on an $m \times n$ matrix $A$, [U,S,V] = svd(A), would cost $O(m^2n + mn^2 + n^3)$ time. But what is the time complexity if we only need the $k$ largest ...
- 327
19
votes
1
answer
494
views
Linear maps between arbitrarily chosen vectors of vector spaces $V$ and $W$
I recently came across this question:
Is the axiom of choice needed to prove the following statement:
Let $V, W$ be vector spaces, and suppose $V \neq \{0\}$. Let $v \in V$, $v \neq 0$, $w \in W$. ...
- 191
7
votes
1
answer
969
views
The height of the Perron-Frobenius eigenvector
Does the height of a real symmetric matrix with non-negative entries control the height of its Perron-Frobenius eigenvector, under some reasonable definition of heights?
Just as an example of what ...
- 23k
5
votes
2
answers
495
views
Existence of parametrizations of rational orthogonal matrices
I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this?
Thanks....
0
votes
0
answers
251
views
Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components
I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
- 11
2
votes
1
answer
520
views
Neighborhood overlap matrix for a bipartite graph
Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...
- 1,068
3
votes
1
answer
414
views
Known Results on Convexity of Numerical Range
Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set
\begin{align}
\mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\}
\end{align}
...
- 1,421
3
votes
1
answer
164
views
A hyperplane inside another one
Let D be a divison ring, let V be a left vector space of over D,
possibly infinite dimensional, and let F be the prime field of D.
Is it true that every F-hyperplane of V contains a D-hyperplane of ...
6
votes
2
answers
346
views
When are two subvarieties of matrices conjugate?
Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then ...
- 101
-1
votes
1
answer
83
views
Dimension of some ideal in the group ring Z/p[Z/p]
Let I be the augmentation ideal of the group ring Z/p[Z/p] and I^n denotes the ideal generated by all possible products of n elements from I.
Question: What is dimension of I^n as a vector subspace ...
- 745
0
votes
0
answers
561
views
What are the properties of this linear operator?
Suppose $f(x)$ is a function which satisfies the following condition:
$$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$
Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...
- 10.1k
3
votes
0
answers
221
views
"Shifted" Vandermonde determinant is nonzero?
I have already posted this question at MSE here, but as it received a few upvotes, but no comments or answers I choose to cross-post it here.
Let $P$ be a degree-two polynomial, with roots $\alpha,\...
- 621
8
votes
1
answer
625
views
A variant of an Eventown problem for modulo a prime number
Consider the following problem, called the 'Eventown problem':
In a town, residents can form different clubs. The town council establishes the following rules:
1) Every club must have an even ...
- 313
4
votes
0
answers
312
views
Dimension of a commuting nilpotent variety
Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
- 768
27
votes
6
answers
6k
views
Origin of exact sequences
I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
- 443
1
vote
0
answers
433
views
Bounding the norm of the Dirichlet kernel as a matrix function
Consider the Dirichlet kerel:
$f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$.
Now, given a diagonalizable real matrix $A$, one can consider $f(A)$, using the standard notation of matrix functions. Namely, $f(A) ...
- 73
0
votes
0
answers
131
views
The largest size of a boolean subgraph (a hypercube) of a given graph
Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...
- 313
7
votes
1
answer
443
views
How much can I perturb a symmetric stochastic matrix and keep positive solutions?
Suppose $A$ is a symmetric stochastic $n \times n$ matrix with least eigenvalue $\lambda_n>0$. Consider real symmetric perturbations $\widehat{A}$.
How large can I take $\epsilon$ such that $\...
- 1,091
3
votes
1
answer
367
views
What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?
If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
- 803
-2
votes
1
answer
213
views
Solving a difficult equation for a variable?
I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
2
votes
0
answers
643
views
Separating the eigenvalues of a Hermitian matrix with a special block structure
I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where,
$J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$
$A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$.
$B \in \...
- 21
4
votes
0
answers
137
views
What do we know about the generalized eigenvalue problem involving a projector?
Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
$$...
- 325
3
votes
1
answer
369
views
Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]
Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n.
We know that $f\equiv 0$. It's call Hausdorff theorem.
This theorem is wrong on $\mathbb{R^+}$, a ...
user46896
0
votes
0
answers
49
views
Forming orthogonal bases in different orders
Let $\alpha_1, \dots, \alpha_n$ be unit vectors in some vector space $V = R^d$. For any permutation $\pi: [n] \rightarrow [n]$, we can form the Gram-Schmidt orthogonal bases $\beta_{\pi,1}, \dots, \...
- 3,475
2
votes
1
answer
2k
views
Matrix equation XAX=B where the solution must be diagonal [closed]
$$X_{solution}=\arg\min_X \|XAX-B|_F \quad\mathrm{subject\ to}$$
X is square and diagonal
A is square and positive semi-definite
B is square and positive semi-definite
Any pointers or relevant ...
- 31
1
vote
0
answers
209
views
An optimization problem on the sphere
Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector $v\in\...
- 13.9k
0
votes
0
answers
104
views
Linear system with many solutions from a finite set
Basically I am looking for a linear system with
many solutions from a finite set.
Choose a finite set of rationals $S$ and fix
positive integer $k$.
Let $A$ be a linear system with $n$ variables $...
- 25.4k
5
votes
2
answers
2k
views
Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix
I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...
- 145
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
1
vote
2
answers
2k
views
Bound on smallest entry of inverse matrix
For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse ...
- 13
26
votes
3
answers
17k
views
Hölder's inequality for matrices
I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...
- 837
3
votes
1
answer
1k
views
Calculating a generalized inverse (Moore–Penrose pseudoinverse)
I am considering a graph with $n$ edges with the following nicely structured adjacency matrix:
\begin{equation}
A_n=
\begin{pmatrix}
0 & 0 & 0 &\cdots & 0 & 0 & 1\\
0 & 0 &...
- 41
1
vote
0
answers
104
views
Dimension of $L(E,F)$
Let $E,F$ be two vector spaces over a field $k$.
$L(E,F)$ is the $k$-vector space of linear maps $E \rightarrow F$.
In ZFC, is there a functionnal relation between $dim(L(E,F))$ and $(dim(E),dim(F),|...
- 2,519
1
vote
2
answers
134
views
Mixing Numerical Range and inner product
Let $\mathbf{A}$ and $\mathbf{b}$ be a symmetric $N\times N$ real matrix and $N\times 1$ real vector respectively. Then consider the set of points in $\mathbb{R}^2$ defined as
\begin{align}
\mathbb{S}=...
- 1,421
2
votes
0
answers
263
views
Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix
I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers.
To be precise, I want ...
- 21
25
votes
1
answer
4k
views
What kind of random matrices have rapidly decaying singular values?
I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
- 118k
15
votes
4
answers
2k
views
More than $n$ approximately orthonormal vectors in $R^n$
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well established that ...
- 1,160
5
votes
0
answers
148
views
Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
- 6,351
5
votes
1
answer
286
views
Are there any known results on numerical ranges of rank-one positive semi-definite matrices?
In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...
- 1,421
4
votes
1
answer
177
views
Finite generation of vector identities
This question is partially motivated by https://mathoverflow.net/questions/158451/looking-for-a-comprehensive-referece-for-vector-identities, although that question may not be appropriate for MO.
...
- 20.5k
6
votes
2
answers
386
views
How to write this result (successive Schur complements compose nicely)
There is an easy theorem in linear algebra that "successive Schur complementations compose nicely": for instance, let's say I have a matrix
$$
M_0=
\begin{bmatrix}
A & B & C \\ D & E & ...
- 20.2k
3
votes
1
answer
264
views
When is this matrix singular?
Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on $\...
- 2,251
2
votes
1
answer
533
views
Radius of the ball where the inverse of Lipschitz maps exists
I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in On the inverse function ...
- 203
1
vote
0
answers
214
views
range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
- 723
8
votes
5
answers
2k
views
Infinite matrix leading eigenvector problem
This question is cross-posted at Math.StackExchange.com.
I'm trying to find the leading eigenvalue and corresponding left and right eigenvectors of the following infinite matrix, for $\lambda>0$:
$$...
- 1,344
21
votes
3
answers
2k
views
Felix Klein on mean value theorem and infinitesimals
This is a reference request prompted by some intriguing comments made by Felix Klein.
In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
- 16.6k
23
votes
5
answers
4k
views
Smallest non-zero eigenvalue of a (0,1) matrix
What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)? I would be interested ...
- 3,377
1
vote
1
answer
321
views
Cycles of Permutation Related to Rectangular Matrix Transposition
let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A_{i,j}=v_{i*m+j}$
Question:
How can the first element ...
- 13.2k
2
votes
1
answer
233
views
Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.
Let $M$ be an $m$-by-$n$ matrix, here are three definitions$^5$ that we could use for rank:
$rk(M) = \min k$ such that for matrices $P$, and $Q$ with $P$ of size $m$-by-$k$ and $Q$ of size $k$-by-$n$ ...
5
votes
1
answer
258
views
Circuits in a linear oriented matroid
Given $E$ a finite subset of a real vector space $V$, a circuit of the associated matroid is a minimal linearly dependent subset of $E$. For each circuit $\underline C$, a minimal linear dependence
$$\...
- 507