**0**

votes

**0**answers

12 views

### Optimal ordering in Jacobi SVD algorithm

In Jacobi SVD algorithm as given here every pair of columns of the matrix is orthogonalized until convergence. I want to know that how does the order of selection of the pair of columns affect the ...

**2**

votes

**0**answers

36 views

### How many unimodular lattices does it take to fill a cube with high probability?

Consider $C_a$ in $\Bbb Z^n$ a cube of height $a$ at origin in positive coordinates with one corner at origin.
Consider the set $M_c$ of all unimodular matrices in $\Bbb Z^{n\times n}$ with each ...

**-4**

votes

**0**answers

30 views

### finding eigenvector [on hold]

I have
where λ1 = λ2 = 6 and λ2 = λ3 = 0.
I wish to find the eigenvectors for these eigenvalues above.
I've tried to turn it into equations and trying to solve them (this is for λ1 & λ2):
...

**0**

votes

**0**answers

10 views

### Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix).
With some ...

**0**

votes

**0**answers

37 views

### SVD alternatives for symmetric matrices

Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices
$$
A = E'E
$$
Practically this can be done easily using SVD ...

**0**

votes

**0**answers

30 views

### max min of ratio of quadratic forms

Consider the optimization over two vectors $x$ and $y$
$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$
for two positive definite matrices $A$ and $B$.
This problem can be ...

**-7**

votes

**0**answers

36 views

### why is $\frac a 0 $ = \infty $? [on hold]

I wish to know the detailed explanation as to why $\frac a 0 = \infty$ ?
I know that in physics such a fraction is meaningless but I will really appreciate mathematical point of view.

**0**

votes

**0**answers

51 views

### Bounding the off-diagonal entries of a hermitian matrix [on hold]

I asked this question on stackexchange and didnt receive any answers, am therefore posting it here.
The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = ...

**0**

votes

**0**answers

74 views

### Orthogonal Procrustes problem for sub-spaces?

By Orthogonal Procrustes problem I mean given matrix $A$ and $B$ finding a orthogonal matrix $R$ which most closely maps $A$ to $B$, this has a solution as shown in ...

**3**

votes

**1**answer

117 views

### A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let ...

**0**

votes

**0**answers

26 views

### Lower bound of general bilinear form [closed]

Suppose I have a bilinear form $X^TAY$ where $X \in R^n, Y \in R^m$ and $Α \in R^{n \times m}$. All elements of $A$ are bounded, that is $\exists \bar a_{ij}>0:|a_{ij}|\le \bar a_{ij}, \forall ...

**0**

votes

**0**answers

129 views

### Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...

**4**

votes

**1**answer

233 views

### Vector with many non-zero coordinates

Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...

**2**

votes

**2**answers

87 views

### Behavior of orbits under small perturbations

Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange.
Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...

**1**

vote

**0**answers

51 views

### Showing positive stability of a matrix constructed from a positive matrix

A is a positive nonsingular matrix. Let $s>\rho(A)$. We want to show that $B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right)$ is a positive stable matrix, i.e., all eigenvalues of this matrix ...

**6**

votes

**4**answers

463 views

### Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago.
For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...

**0**

votes

**0**answers

48 views

### a two dimensional integer bijection

I want to try to find a specific two-dimensional linear integer bijection. This is to be used in a double sum rearrangement. It's kind of complicated, but I would really appreciate if anybody has any ...

**1**

vote

**0**answers

73 views

### How to write a braiding as a matrix?

Let $V$ be the vector representation of $sl_n$. Then $V \otimes V$ is a $U_q(sl_n)$-module. Suppose that a braiding
\begin{align}
\Psi: V \otimes V \to V \otimes V
\end{align}
satisfies the ...

**4**

votes

**1**answer

106 views

### About the Eigenvalues of Orthogonal Matrix plus Perturbation

Let $O$ be an orthogonal matrix, $O^T O = I$, thus its eigenvalues lie on the unit circle, $\lambda(O)=e^{i\theta}$. Furthermore, assume the form
$O = X Y$, where both matrices satisfy $X^2 = I$ and ...

**1**

vote

**0**answers

62 views

### Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...

**4**

votes

**1**answer

89 views

### Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge ...

**1**

vote

**1**answer

331 views

### Integer Polynomial solutions to functional equation

Recently I came across a functional equation which always has a polynomial with integer coefficients solution.
Let
$$
L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i).
$$
Problem:
...

**0**

votes

**1**answer

70 views

### Finding a vector representation for a data where we only know the inner products

I am an engineer working on speech signal processing and I have a problem that I have encountered while trying to model speech signals. The mathematical formulation is not entirely pure and I try to ...

**1**

vote

**0**answers

62 views

### coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant ...

**5**

votes

**2**answers

281 views

### Explicit solution to a Rayleigh quotient equation

For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
Main equation:
$$
...

**1**

vote

**0**answers

29 views

### Orthogonal basis from corresponding points in different unknown basis [closed]

I have measurements of two points from different unknown basis (A and B). All I know is that the point in A correspondings to the point in B (by corresponds I mean they originate from the same point ...

**4**

votes

**1**answer

72 views

### Perturbations on the pseudoinverse of a matrix

Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation
$$
A_p = A + \Delta
$$
is there a way to represent
$$
(A_p)^{\star}= (A)^{\star} + f(\Delta)
$$
where $(A_p)^{\star}$ ...

**0**

votes

**1**answer

81 views

### Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...

**5**

votes

**0**answers

141 views

### If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?

$\newcommand{\Vect}{\mathsf{Vect}}
\newcommand{\nats}{\mathbb{N}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Alg}{\mathsf{Alg}}
\newcommand{\CAlg}{\mathsf{CAlg}}
\newcommand{\Hom}{\mathrm{Hom}}$
Let ...

**4**

votes

**1**answer

93 views

### Behaviour of eigenspaces of adjacency matrices after a single change to the graph

Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...

**3**

votes

**0**answers

84 views

### Finding nearest Toeplitz matrix to a given matrix

For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz ...

**3**

votes

**1**answer

129 views

### Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...

**3**

votes

**0**answers

105 views

### An $\mathsf{SL(n,F)}$ decomposition problem

Given $n\times n$ square matrix $A\in\Bbb F^{n\times n}_{}$ where $\Bbb F$ is a field is there an easy way to test there is NO decomposition $A=B+C+D$ where $B,C,D\in\mathsf{SL}(n,{\Bbb F})$ are ...

**1**

vote

**0**answers

11 views

### non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...

**1**

vote

**1**answer

82 views

### Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that ...

**-2**

votes

**1**answer

95 views

### Eigenvalues of cyclic tridiagonal matrix [closed]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that this matrix ...

**4**

votes

**1**answer

175 views

### Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...

**2**

votes

**1**answer

78 views

### Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...

**1**

vote

**1**answer

89 views

### Size of connected components of a graph via its spectrum

I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is:
Is there a way to understand the size of each connected ...

**3**

votes

**0**answers

76 views

### Number of linearly independent subsets in arbitrary set

Consider $n$-dimensional vector space $\mathbb{F}_p^n$ over finite field $\mathbb{F}_p$ and a function $a: \mathbb{F}_p^n\rightarrow\mathbb{C}$. Let
$$ F(a) = \sum_{\substack{S\subset \mathbb{F}_p^n ...

**2**

votes

**0**answers

75 views

### Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane

Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i $. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix ...

**1**

vote

**1**answer

41 views

### range of singular values of sub-matrices

Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and ...

**2**

votes

**0**answers

102 views

### Can the matrix exponential be equal to the elementwise exponential [closed]

Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential ...

**1**

vote

**3**answers

163 views

### The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra

Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X)
\otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...

**1**

vote

**1**answer

111 views

### Modified interlacing of eigenvalues

Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A ...

**-2**

votes

**1**answer

74 views

### Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...

**3**

votes

**1**answer

187 views

### Recursively calculate the determinant

A generic $k \times k$ block symmetric matrix $\Sigma$ is denoted as
\begin{align}
\Sigma = \begin{bmatrix}\Sigma_{11} & \Sigma_{12} & \ldots & \Sigma_{1k} \\ \Sigma_{21} & \Sigma_{22} ...

**1**

vote

**0**answers

34 views

### Matrix diagonalization after a completely positive transformation

I have a hermitian matrix $A$ which can be diagonalized:
$$A=UDU^+,$$
where U is the unitary matrix and D is the diagonal matrix.
Next, I have a completely positive transformation over it, which is ...

**1**

vote

**1**answer

62 views

### Spectral radius's relation with row sum

Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$.
I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...

**1**

vote

**0**answers

37 views

### The application of recursive SVD [closed]

Given an n*m matrix A, the SVD decomposition of A is ${\rm SVD}(A)$= $USV^t$.
The application of SVD to the product of U and S gives as a result the same matrices multiplied by the identity matrix, ...