Questions about the properties of vector spaces and linear transformations, including linear systems in general.
4
votes
1answer
81 views
What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$
The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
-1
votes
0answers
11 views
regu tools l_curve regularization stanford ee 263 [migrated]
I am trying to solve one of the famous stanford EE263 problems, which gives me matrix A representing blurring of an image and y, representing the blurred image. For that I have been trying to use ...
2
votes
0answers
58 views
Infinite series of determinants
I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...
-2
votes
1answer
35 views
How to find a matrix by its characteristic value and characteristic vectors? [on hold]
Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen vlaues) and characteristic vectors (eigen vectors). But my qustion is why cant we ...
1
vote
0answers
29 views
Lanczos algorithm with thick restart on a dynamic matrix
currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...
0
votes
0answers
66 views
Eigenvalue of (0-1) matrix
Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10)
The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
8
votes
0answers
187 views
Lift chain complex from $F_2$ to $Z$
We start with a finite dimensional chain complex over $F_2$, equipped with a basis. That is, we have finitely many finite dimensional $F_2$-vector spaces $C_0,\dots,C_k$ with bases $B_0,\dots,B_k$, ...
0
votes
0answers
30 views
Bounding Rayleigh quotioent for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
2
votes
1answer
105 views
About partial uniqueness of SVD
In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let ...
1
vote
0answers
40 views
For Finite Dual when is $(A \otimes A)^o = A^o \otimes A^0$?
Let $A$ be any $k$-algebra. The finite dual or restricted dual of $A$ is
$$
A^o = \{f \in A^* ~ | ~ f(I)= 0, \text{ for some ideal } I \subseteq A, \text{ such that } \text{dim}_k(A/I) < \infty\}.
...
2
votes
0answers
35 views
successive schur complements
If I have a large (e.g. 6000x6000), sparse, positive definite matrix $M$ (which may have individual entries everywhere, but most non-zero entries are on / around the diagional).
Divide $M$ into blocks ...
3
votes
0answers
120 views
Symmetric matrices with $\rho(A)\gg\|A\|_\infty$
For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
0
votes
0answers
37 views
Norm bound of a complex resolvent
A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{ii}|-\sum_{j \neq i}|a_{ij}|}$, where the ...
3
votes
0answers
28 views
Simultaneous Tridiagonalization of a given set of hermitian matrices?
I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?
4
votes
2answers
229 views
How to check whether a matrix is completely positive or not?
The definition:
cone of completely positive matrices
$\mathcal{C}=\{\sum_{i=1}^kx_ix_i^T:x_i\in\mathbb{R}^n_+\ for \ i=1,2,...,k\}$.
I just don't knwo how to check whether a matrix belongs to ...
3
votes
1answer
39 views
submatrix of a given size with maximum frobenius norm
Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
0
votes
0answers
33 views
Representing product and sum updates of values with matrix difference equations [closed]
A vector v updates all of its elements simultaneously in discrete time steps. At each time step, each element of v may change in one of the following ways:
It may remain the same. (ex: v1 = v1)
It ...
-1
votes
0answers
42 views
Feature relationship based class separability [closed]
I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description.
I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
7
votes
0answers
84 views
When is a product of hyperbolic matrices hyperbolic?
Suppose $A_1,\ldots,A_n$ is a sequence of $2 \times 2$ complex matrices such that $| \det(A_j) | =1$ and $ | \mathrm{tr}(A_j) | > 2 $ for each $j$. What kinds of reasonable restrictions can one ...
1
vote
0answers
85 views
Distributing the Hodge map over the wedge product
Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...
0
votes
0answers
31 views
Compare full-rank probabilities of products of random matrices
Consider two matrices $C_1=A\times B_1$ and $C_2=A\times B_2$, where $A\in\mathbb{F}_q^{N\times K}$, $B_1\in\mathbb{F}_q^{K\times M}$ and $B_2\in\mathbb{F}_2^{K\times M}$; $M\leq N\leq K$.
It is ...
0
votes
0answers
74 views
Complexity of turning a d-degree polynomial to 2-degree polynomial
For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
-2
votes
0answers
19 views
how to complete arbitrary basis knowing 2 orthonormal vectors of Rd (d > 2) [migrated]
In a paper the following statement is used: "To construct the matrix B, complete the vectors (y, x) to an arbitrary basis of Rd and then apply the Gram-Schmidt orthonormalisation".
assume we know x ...
14
votes
0answers
275 views
a naive question about the second dual of a vector space
Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...
3
votes
0answers
135 views
Is there such a matrix in $SO(n)$?
Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = ...
5
votes
4answers
259 views
NP-hard problems in linear algebra and real analysis [closed]
I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...
3
votes
3answers
190 views
Square root of a complex matrix commuting with a given one
Assume two commuting $n\times n$ complex matrices $A$ and $B$ are given. Then it is in general false that if $C$ is a square root of $A$, i.e., if $C^2=A$, then $C$ commutes with $B$ (the simplest ...
-4
votes
0answers
20 views
Relationship for cosine of angle [migrated]
If $x$ is the cosine of the angle between the vectors $a$ and $b$, $y$ is the cosine of the angle between the vectors $a$ and $p$, and $z$ is the cosine of the angle between the vectors $b$ and $p$, ...
0
votes
1answer
57 views
dual space of the quotient space of some locally convex topological space
I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
1
vote
0answers
56 views
What is the time complexity of approximated SVD
Full SVD, on an m*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 singular values, say, $[U_k,S_k,V_k] = ...
15
votes
1answer
232 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 $v \in V$, $v \neq 0$, $w \in W$. ...
3
votes
1answer
178 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 ...
2
votes
1answer
77 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?
...
0
votes
0answers
55 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 ...
1
vote
1answer
72 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 $i,j \in V$, $$a_{ij} = |N_i \cap ...
2
votes
1answer
65 views
Known Results on Convexity of Numerical Range
Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set
\begin{align}
\mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\}
\end{align}
...
3
votes
1answer
142 views
A hyperplane inside another one
Let D be a divison ring, let V be a left vector space of over D,
possibly infinite dimensional, and let F be the prime field of D.
Is it true that every F-hyperplane of V contains a D-hyperplane of ...
6
votes
2answers
284 views
When are two subvarieties of matrices conjugate?
Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then ...
-1
votes
1answer
58 views
Dimension of some ideal in the group ring Z/p[Z/p]
Let I be the augmentation ideal of the group ring Z/p[Z/p] and I^n denotes the ideal generated by all possible products of n elements from I.
Question: What is dimension of I^n as a vector subspace ...
0
votes
0answers
80 views
What are the properties of this linear operator?
Suppose $f(x)$ is a function which satisfies the following condition:
$$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$
Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...
3
votes
0answers
85 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 ...
-1
votes
0answers
29 views
Is it possible to transform projections? [migrated]
From linear algebra I know how to transform a point or a direction from one space to another using transformation matrices (and I think it's pretty awesome by the way). But what about projections? Is ...
6
votes
0answers
120 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 ...
3
votes
0answers
129 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 ...
15
votes
5answers
2k 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 ...
0
votes
0answers
67 views
Bounding the norm of the Dirichlet kernel as a matrix function
I asked the following there also, but I am beginning to think it would be more appropriate here.
Consider the Dirichlet kerel:
$f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$.
Now, given a diagonalizable real ...
0
votes
0answers
74 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 ...
7
votes
1answer
129 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 ...
3
votes
1answer
182 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, ...
-2
votes
1answer
121 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 $A, \beta, r, d,$ ...
