**3**

votes

**0**answers

34 views

### Adjacency matrix, quivers

Let $Q$ be a quiver with finitely many edges and such that the underlying graph is connected. Let $I = \{1, \dots, n\}$ be the vertex set of $Q$, so we have $\mathbb{R}\{I\} \cong \mathbb{R}^n$.
For ...

**0**

votes

**2**answers

42 views

### Solving sparse linear least squares or a positive definite 5-band matrix system fast

I want to quickly solve linear least squares problem for $x \in \mathbb{R}^n$
$$min_x \left\| A x - b \right\|_2^2$$
with a special sparse structure where each row in $A$ has only up to 4 ...

**-4**

votes

**0**answers

23 views

### constant rate of change [on hold]

When downloading a large file, Travis noticed that the estimated time remaining to complete the download decreased by 35 seconds for each additional megabyte downloaded. When he started the download ...

**0**

votes

**1**answer

36 views

### The Condition Number of a Scaled Vandermonde

Let $V(x_1,..,x_n)$ be the Vandermonde induced by $x_1,..,x_n$ and
Let $\tilde{V} = V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant to such ...

**3**

votes

**2**answers

74 views

### Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...

**2**

votes

**1**answer

53 views

### Set of density matrices

A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$
In Quantum Mechanics it is natural to look at a group action
$\Phi: ...

**-4**

votes

**0**answers

40 views

### Algebra II exercise (help) [on hold]

Can anyone please help me with this exercise for my exam?
It says:
1)Given S={(x₁,x₂,x₃) ∈ R^3 : x₁ + x₂ -2x₃ = 0}.
a)Prove that S is a subspace.
b)For each of the matrices A shown, check if ...

**15**

votes

**2**answers

788 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**53**

votes

**11**answers

6k views

### Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...

**2**

votes

**0**answers

48 views

### Polynomials with positive coefficients passing through fixed points/range of Vandermonde matrices

I'll give two equivalent statements of the setup, then give my questions.
Fix integers $M \leq N$ and define the Vandermonde-like matrix $V_{M,N}[i,j] = (1 - \frac{i}{M})^{j-1}$ for $i \in ...

**-3**

votes

**0**answers

38 views

### eigenvalues of cycle and its complement [on hold]

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...

**-4**

votes

**0**answers

26 views

### What is the upper bound for training Linear separable set with Perceptron, Rosenblatt rule? [on hold]

I have the following neural networks problem and couldnt find any answer on the web. Any hints would help. I am not looking for a complete solution, just some pointing in the right direction.
...

**3**

votes

**1**answer

231 views

### Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...

**-1**

votes

**0**answers

37 views

### A question on spectrum [migrated]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$.
Suppose
$M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$
$F(A) = ...

**-3**

votes

**0**answers

40 views

### Decomposition of orthogonal matrix into 2 orthogonal matrices [on hold]

Is there anyway to find a decomposition of orthogonal matrix $A$ into 2 orthogonal matrices $P$ and $Q$ such that $A = PQ^T$?

**-1**

votes

**0**answers

56 views

### Interpolating Product of two Polynomials

Consider we have two non-constant polynomials $A(x)$ and $B(x)$. We define the polynomials over field $\mathbb{Z}_p$, for a large prime number $p$.
We define Polynomial $A(x)$ as follows: ...

**1**

vote

**0**answers

63 views

### Interpolating a Polynomial Given Multiplier of each $y_i$

It would be great even if you answer only one of the below questions.
We have polynomial $P(x)=(x-\beta)\cdot g(x)$, where degree of $P(x)$ is fixed n-1, $\beta$ chosen uniformly at random from the ...

**0**

votes

**2**answers

95 views

### Estimating the shift in the $\lambda_{max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...

**3**

votes

**2**answers

92 views

### Spectral theorem from Jordan decomposition in infinite dimensions

The finite-dimensional spectral theorem is a simple special case of the finite-dimensional Jordan decomposition. We also have various infinite-dimensional generalizations of the spectral theorem; can ...

**0**

votes

**1**answer

150 views

### Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...

**0**

votes

**0**answers

27 views

### Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...

**2**

votes

**1**answer

205 views

### Two Concepts of Monotonicity

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that
$F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that
$$
\langle F(y)-F(x), ...

**4**

votes

**0**answers

40 views

### $AXB$ sort of decomposition? [migrated]

Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1 \dots, B_d ...

**6**

votes

**1**answer

157 views

### Computation Time of Smith Normal Form in Maple

I am using maple to compute the Smith Normal Form of a matrix of size 120*120 and it seems that I will never get an answer back. I have checked my code for small cases and I believe that it is ...

**4**

votes

**0**answers

150 views

### Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...

**3**

votes

**1**answer

101 views

### Number of linearly bisected subsets in finite vector space $F_2^n$

We consider the $n$-dimenstional finite vector space $\mathbb{F}_2^{n}$ over the finite field of two elements. For a subset $A\subseteq \mathbb{F}_2^{n}$ of even size $|A|=2m$ and a linear form ...

**-5**

votes

**0**answers

40 views

### Function that outputs only 1 or 0 depending on sign of variable? [closed]

Is there a single variable (preferably simple) function which equals 0 for any positive input and 1 for any negative input, or vice versa?

**5**

votes

**1**answer

272 views

### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...

**1**

vote

**0**answers

48 views

### Avoiding the range of a bivariate integer function or Diophantine function [closed]

I have a bivariate integer function $f(x,y)=5+23x+7y+30xy$ where $x,y \geq 0$ and are integers. The lattice points of this function, or its range contain a large number of values. I'm trying to see if ...

**3**

votes

**1**answer

155 views

### On the search for an explicit form of a particular integral

Let $f$ be integrable over the interval $(0, 1)$, and
$$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$
Suppose $f(x) = f(1-x)$; we can then show that
$$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...

**2**

votes

**1**answer

88 views

### Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...

**4**

votes

**0**answers

86 views

### An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132)
For the argument to ...

**-3**

votes

**0**answers

46 views

### How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$ [closed]

How to find $B$ by solving the following linear system:
$s_k$ $B$ ${s_k}^T$ $=1,$ $\qquad$ for $k=1 ... ,p$.
Where $s_k$ is a $1\times3$ row_vector from the matrix
$S= [s_1 ... ...

**2**

votes

**1**answer

163 views

### About distinct eigenvalues of a graph

if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that ...

**6**

votes

**2**answers

865 views

### Computation of a Drazin inverse

I need to compute the Drazin inverse $A^D$ of a singular M-matrix $A$, i.e., a matrix in the form $A=\lambda I -P$, where $P$ has nonnegative entries and $\lambda$ is the spectral radius (Perron ...

**5**

votes

**0**answers

100 views

### Biggest (or large) rectangle in a polytope

I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...

**0**

votes

**1**answer

327 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 ...

**0**

votes

**0**answers

67 views

### A question in compact set

Definitions:
${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$
${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...

**6**

votes

**0**answers

151 views

### When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...

**0**

votes

**0**answers

46 views

### Equivalence classes of pairs linear transformations

Consider the set of 4-tuples:
$$S_{(x, y), k} = \{ (a_i, b_i, a_j, b_j) : \|a_ixb_i - a_jyb_j\|_F^2 = k \}$$
for $a \in GL(m, \mathbb{R})$, $b \in GL(n, \mathbb{R})$, $x, y \in \mathbb{R}^{m \times ...

**1**

vote

**1**answer

87 views

### Transversality in Morse theory, linear algebra version

I am working on a product in Morse-Bott homology which has led me to the following considerations and unanswered question. I would be very grateful if anyone could help.
Suppose $H:\mathbb{R}^n \to ...

**15**

votes

**2**answers

502 views

### Matrix equation $XAXBXC=I$

Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution ...

**0**

votes

**1**answer

245 views

### Comparing iterative methods for linear systems

For a tridiagonal matrix of the from
\begin{bmatrix}
a & -b & \newline
-b & a & -b \newline
& \ddots & \ddots & \ddots \newline
& & & & ...

**4**

votes

**3**answers

134 views

### Is an associative division algebra required for this phenomenon?

For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...

**3**

votes

**1**answer

250 views

### Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$,
the group of invertible upper triangular $n\times n$ matrices.
I know that if $\rho : G\rightarrow T(n,k)$ is faithful
(i.e. into) then ...

**2**

votes

**0**answers

187 views

### Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way.
All the ${\lambda}_i$ are distributed the same way with chi-square ...

**0**

votes

**1**answer

82 views

### Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: ...

**0**

votes

**1**answer

58 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...

**-1**

votes

**0**answers

47 views

### Orbit closures of symmetric bilinear form [migrated]

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where:
...

**2**

votes

**1**answer

256 views

### Is there an algorithm to test whether a vector is an eigenvector of a power of a matrix?

Given a square matrix $A\in k^{n\times n}$ and a vector $x\in k^n$ over some field $k$, is there an algorithm to test whether there are $s\in\mathbb{N}$ and $\lambda\in k$ such that $A^sx=\lambda x$? ...