Questions about the properties of vector spaces and linear transformations, including linear systems in general.
-3
votes
0answers
23 views
Basis of a vector space $V$ [on hold]
If $S_{i}$ is a subset of linearly independent vectors of a vector space $V$, and $S_{g}$ a subset of generators of $V$. Are there $S_{g}^{'}\subset S_{g}$ with $S_{i}\cup S_{g}^{'}$ a basis of $V$?
3
votes
0answers
103 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 ...
12
votes
5answers
1k 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
15 views
What book or (math)course do you recommend? [migrated]
Can anyone recommend me a book or an online math course, that can help me to overcome my fear in math? I'm sick and tired of being afraid of numbers.
Thanks in advance!
0
votes
0answers
50 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 ...
-2
votes
0answers
65 views
Tensor products [on hold]
I'm currently trying to teach myself about tensors but I'm having some trouble with understanding what's going on.
I've managed to come to a basic understanding of the ranks of tensors, rank n ...
0
votes
0answers
63 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
113 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
vote
0answers
35 views
Invertibility of NxN nonnegative matrix with diagonally dominant elements [migrated]
I have a NxN nonnegative matrix where the diagonal element of any row i is greater than the off diagonal elements, 1 > aii > aij ≥ 0 for j not equal to i . This not a diagonally dominant matrix as it ...
3
votes
1answer
171 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
113 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 $A, \beta, r, d,$ ...
1
vote
0answers
76 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 ...
0
votes
0answers
29 views
problem computing block inverse of factorized matix [migrated]
I'm running into trouble when verifying the inverse of the following block matrix using the Schur complement:
My matrix is given by:
$$K = USU^T,$$
where U are its eigenvectors and S the ...
2
votes
0answers
41 views
What do we know about the generlized 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
...
0
votes
0answers
35 views
Eigenvalues of a “Half-Kronecker ” Product
The Problem:
Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other
2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given).
Now we can construct a new matrix ...
0
votes
0answers
40 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, ...
2
votes
1answer
127 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 ...
0
votes
0answers
37 views
Generalized Eigenvalue problem solution
$\circ$ Consider the following eigenvalue problem : $$XY^{T}YX^{T}w=\lambda XX^{T}w \hspace{0.5cm} (1)$$
Matrices $XY^{T}$ and $XX^{T}$ $\in \mathbb{R}_{n \times n}$ are positive semi-definite and ...
1
vote
0answers
151 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 ...
0
votes
0answers
79 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 ...
4
votes
2answers
133 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:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...
-1
votes
0answers
16 views
This is regarding a verbal math problem [migrated]
A Shopkeeper buys sweets at 11 for a rupee. He bought an equal number of sweets at 10 for a rupee. he sold all the sweets at 8 for a rupee. find loss or gain %
Please solve this for me. I am not able ...
3
votes
1answer
91 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 ...
1
vote
2answers
74 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 ...
7
votes
1answer
133 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 ...
3
votes
1answer
107 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 ...
0
votes
0answers
85 views
Extending Continuous Basis [migrated]
It is given $(k-d)$ continuous vector-valued functions $K_1,\dots,K_{k-d}:\mathbb{R}\mapsto\mathbb{R}^k$, with $d\leq k$. Suppose that for all $x\in\mathbb{R}^k$, the set ${\cal ...
1
vote
0answers
75 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 ...
1
vote
2answers
91 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}
...
1
vote
0answers
80 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 ...
4
votes
0answers
70 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 ...
5
votes
1answer
82 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 ...
4
votes
1answer
118 views
Finite generation of vector identities
This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO.
Consider the set $\mathcal{E}$ of all valid ...
6
votes
2answers
122 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 & ...
3
votes
1answer
172 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 ...
0
votes
0answers
29 views
SDP formulation of noisy low rank matrix completion (2) [migrated]
Thank you Michael for the answering my previous question, SDP formulation of noisy low rank matrix completion. It seems that I overlooked the problems in my initial question. I didn't recognize the ...
0
votes
0answers
24 views
Reference Request: Matrix of a composition(sum) of two operators is the Kronecker product (sum) of the matrices of each operator [migrated]
Here is link to an example: http://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians, but it provides no references.
Hoffman and Kunze(Linear Algebra) develop linear algebra with explicit ...
0
votes
0answers
49 views
Linear Algebra: spectral decomposition by special single row replacement
Problems Statement:
Let $\mathbf{Y}$ be an $m \times n$ real matrix whose $i$th row is $\mathbf{y}_i = \left(y_{i1},...,y_{in}\right)$ and whose rank is $n$. Let $\mathbf{y}_{i,\sigma
}=\left( ...
1
vote
0answers
55 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 ...
0
votes
0answers
12 views
Weighted linear regression [migrated]
I have a set of n events. Each event has m variables. At least 1 event produces an observation. It is possible for several events to occur simultaneously.
E.g.
4 Events. 3 variables each:
E11=0.4, ...
4
votes
3answers
171 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 $\lambda>0$:
...
-3
votes
0answers
22 views
formula for computing inverse of a matrix [migrated]
I am pretty sure I once heard of a formula of computing the inverse of a matrix $I-A$ based on the inverse of $A$, but I can't remember the name of the formula. Is there a shortcut for computing the ...
-3
votes
0answers
19 views
Is there a way to find the SVD in the following case? [migrated]
Say we have a matrix $A$ with SVD $A = U \Sigma V'$.
Now say we multiply each column of $A$ by a scalar (the scalar can differ across columns).
Is there a way to find the SVD of the new matrix based ...
14
votes
4answers
502 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 ...
0
votes
0answers
45 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 ...
2
votes
1answer
16 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 ...
-1
votes
0answers
15 views
Rank of matrix with parameters [migrated]
i need a little help with an exercise on my class. We have to do this exercise
for a = 2, b = 4, c = 0
How can i determine λ if a know that the rank of the matrix is 2 ?
3
votes
1answer
68 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
...
-1
votes
0answers
33 views
Minimum Eigenvalue of the Rank One update to a Positive Semi-Definite matrix [migrated]
Let $\mathbf{A}$ be a $N\times N$ positive semi-definite hermitian matrix. Let $\mathbf{b}$ be a $N\times 1$ complex vector. For any given constant $t$, I interested in the minimum eigenvalue of the ...
-3
votes
0answers
14 views
Movement of a point at the and of a rotating line problem [migrated]
I've got a maths problem I'm hoping someone can help me with.
Please see the image below.
Point A (top left hand corner of the rectangle) exists at an x/y coordinate.
If I rotate the rectangle 45 ...
