Questions about the properties of vector spaces and linear transformations, including linear systems in general.
0
votes
0answers
25 views
problem computing block inverse of factorized matix
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
26 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
27 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
36 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
108 views
Matrix equation XAX=B where the solution must be diagonal [on hold]
$$X_{solution}=\arg\min_X \|XAX-B|_F \quad\mathrm{subject\ to}$$
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
31 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 ...
-4
votes
0answers
51 views
Derivative of diag (A) with respect to A [on hold]
Assume matrix A has no special structure, how to calculate
?
Where means getting the element from the diagonal. I'm not sure whether $$\delta(A)$$ is a column vector consisting the diagonal element ...
1
vote
0answers
148 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
76 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 ...
3
votes
2answers
100 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
88 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
73 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
123 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
100 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
73 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
88 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
78 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
69 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
76 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
114 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
170 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
487 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
42 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
14 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
67 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 ...
6
votes
1answer
231 views
Injectivity of matrix “fingerprint”
Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries.
For any matrix $A$, define
$$ ...
0
votes
0answers
47 views
A lower bound on the number of matrices whose image contains all multiples of $p^e$
Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that ...
3
votes
0answers
58 views
Orthonormal basis with small $\ell_{\infty}$ norm for a subspace
(Note: I posted this question on stackexchange but was told it might be more suited for mathoverflow. There has been some discussion about the problem there.)
Let $U \subset \mathbb{R}^n$ be a ...
1
vote
0answers
96 views
Closed-form solution to a system of linear equations
Consider the following $n \times n$ matrix with a particularly nice structure:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 &1\\
0 & 0& \dots&0 & ...
6
votes
1answer
151 views
Almost Hadamard matrices
As well-known, a Hadamard matrix is a square matrix with all coefficients $\pm 1$
and pairwise orthogonal rows or columns. Such matrices exist conjecturally
in every dimension divisible by $4$. Call ...
0
votes
1answer
95 views
Find the following transformation $G$
I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.
I'm ...
10
votes
0answers
213 views
Who stated and proved the “Hopf lemma” on bilinear maps?
If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.
Nondegenerate here means ...
2
votes
0answers
49 views
Can sparse matrices satisfy the Null Space Property?
Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if
$$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ...
3
votes
1answer
123 views
Reconstructing a (unitary) matrix from the determinant of its sub-matrices
I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ ...
0
votes
1answer
119 views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...
1
vote
0answers
139 views
Incoherence of the row/column span
Due to V.Chandrasekaran., et al (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that:
$$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$
where the lower bound is achieved (for ...
