**0**

votes

**1**answer

38 views

### The effect of linear transformation on generic vectors [closed]

I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$.
I know that if $M$ has p-norm $\|M\|_p = \lambda$, then by ...

**3**

votes

**1**answer

105 views

### Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition
Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are ...

**2**

votes

**1**answer

109 views

### Commuting nilpotent matrix collection

For every large enough $m\in\Bbb N$ are there $c=\alpha m$ (for some fixed $\alpha>0$) square matrices $A_1,\dots,A_c$ that commute with each other with nonzero product ($\forall ...

**8**

votes

**1**answer

126 views

### Exact eigenvalues of a specific tridiagonal matrix

I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...

**4**

votes

**1**answer

119 views

### Representing a graph's vertices as linear combinations of paths

I have a (directed graded) graph $G=(V,E)$ with two distinguished subsets of $V$: sources and destinations. I am to show that the set of indicator functions of paths starting in a source and ending in ...

**0**

votes

**0**answers

175 views

### A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration
$\hat{c} = f(c)$, where each element of $\hat{c}$ is given by
$$\hat{c}_{\ell} = \frac{\sum_k ...

**9**

votes

**1**answer

223 views

### How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently

Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply ...

**3**

votes

**0**answers

58 views

### How to construct large sets of $m$-dimensional vectors over a finite field such that any $m$ of them are independent?

Let $m\ge 2$ be an integer and $\mathbb{F}$ be a finite field of order $p^k$. I want to construct as many as possible $m$-dimensional vectors $v_1,v_2,\ldots,v_n$ in field $\mathbb{F}$ such that any ...

**2**

votes

**0**answers

75 views

### Quantum Groups and quantum spaces - From algebra to Analysis

My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...

**1**

vote

**2**answers

99 views

### Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...

**0**

votes

**0**answers

47 views

### Separable Least squares - is there a notion of conjugate directions?

I have a general question.
Suppose I have the following to optimize
$$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$
where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...

**9**

votes

**2**answers

478 views

### Do smooth manifolds admit linear atlases? [duplicate]

There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique).
I am ...

**2**

votes

**2**answers

434 views

### For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?

Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$?
In particular, what if $R$ is the group ring ...

**1**

vote

**1**answer

326 views

### Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...

**0**

votes

**1**answer

77 views

### Large Tridiagonal Matrix - Eigenvalues

Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):
$M =
\begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
...

**0**

votes

**1**answer

154 views

### $P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and
$P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable.
$Z$ is eigenvalue of $P(Z )$ if ...

**1**

vote

**1**answer

43 views

### On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...

**7**

votes

**1**answer

134 views

### Can every trace preserving isomorphism of unital self-adjoint matrix algebras be realized as conjugation by a unitary?

In this paper, Friedland shows (in Lemma 3.4) that if $\phi$ is an isomorphism of coherent algebras, then there exists a unitary $U$ such that
$$ \phi(M) = UMU^\dagger$$
for all $M$. I am wondering if ...

**4**

votes

**1**answer

289 views

### Unusual inequality concerning elementary symmetric functions

To state the question and fix conventions I will introduce some notation from e.g. (Lin-Trudinger, Bull. Aust. Math. Soc. 1994, ``On some inequalities for elementary symmetric functions")
Given ...

**2**

votes

**0**answers

107 views

### Eigenvalue perturbation of a symmetric matrix by a random orthogonal projection

Given fixed real symmetric $D\in\mathbb{R}^{n\times n}$ with $n$ distinct eigenvalues, let $U$ be a random orthogonal matrix selected uniformly from the space of $n\times n$ orthogonal matrices, and ...

**-1**

votes

**2**answers

173 views

### Are the coefficients of a linear combination of random vectors as random?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...

**5**

votes

**2**answers

158 views

### The coefficient of a specific monomial in the expansion of the following polynomial

Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq ...

**3**

votes

**1**answer

42 views

### Complexity class of matrix generalization of knapsack problem

Let $n$ be a natural number, $u_+,v_+,u_-,v_-$ be real or complex column vectors of length $n$, and $M_1,M_2,\ldots,M_k$ be a finite collection of $n\times n$ real or complex matrices.
Consider the ...

**11**

votes

**2**answers

280 views

### A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**0**

votes

**1**answer

71 views

### a sum of ratios of quadratic forms

I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} ...

**0**

votes

**0**answers

43 views

### Solving a system of quadratic equations over a subspace

Let $A \subseteq \mathbb{R}^{n}$ be subspace of dimension $d$, parametrized as $A=\{x|Vx=0\}$, where $V$ is a suitable $d \times n$ matrix.
Now a system of $m$ quadratic equations ...

**1**

vote

**1**answer

60 views

### Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...

**1**

vote

**0**answers

39 views

### Complex conjugate and unitary complex conjugate

Definition: Let V be complex finite dimensional inner product space
Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex ...

**4**

votes

**1**answer

143 views

### Ideals in exterior algebras over the field with two elements

Suppose we have an exterior algebra over $\mathbb{F_2}$, say $R = \Lambda_{\mathbb{F_2}}V$, where $V$ is an $n$-dimensional $\mathbb{F}_2$ vectorspace. Let $x_1,\ldots,x_n$ be a basis of that ...

**3**

votes

**1**answer

126 views

### Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j ...

**1**

vote

**1**answer

30 views

### Similarity transform of a diagonalizable matrix that minimizes the Euclidean condition number

If I have a diagonalizable matrix $A = V\Lambda V^{-1}$, is there a way to show that for any similar $B$ such that $B = T\Lambda T^{-1}$, the Euclidean condition number $\kappa_2(B) \geq ...

**0**

votes

**1**answer

264 views

### How do Schubert classes form a basis for $H^{*}(Gr(k, n))$?

I've gone through many texts in algebraic geometry, specifically, Schubert calculus. They all claim that the Schubert classes $[\Omega_{\lambda}]$ form a basis for the cohomology ring of the complex ...

**2**

votes

**0**answers

407 views

### On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) \subseteq TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. Let $\bar{J} : TTM \longrightarrow ...

**3**

votes

**2**answers

259 views

### A question on determinant of a matrix polynomial

Let
${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.
${\rm{P(}}\lambda {\rm{) = ...

**35**

votes

**13**answers

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**20**

votes

**0**answers

566 views

### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...

**4**

votes

**0**answers

105 views

### Hodge duality and the determinant of the product of two matrices

I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are ...

**2**

votes

**0**answers

51 views

### Determining Inconsistency of (first-order) Non-linear System of Equations [closed]

Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?
Take the following system of equations as an example. The ...

**2**

votes

**0**answers

62 views

### When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?

It would be soo awesome if you could help!
For $ p \in \mathbb{N}$ consider the following $\mathcal{S(H)}^{p\times p}$-matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p}$ of nuclear operators ...

**0**

votes

**0**answers

23 views

### On a certain notion of sequence rank

Given a pair of length $n$ vectors $(a,b)$ denote $a\oplus b$ to be the operation that yields an $n\times n$ matrix with entry $i,j$ given by $a_i+b_j$ and every diagonal entry $0$
Given a zero ...

**2**

votes

**1**answer

99 views

### A graph assignment problem

Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in ...

**2**

votes

**0**answers

127 views

### On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
...

**0**

votes

**1**answer

95 views

### Fixed point of quantum operations

A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...

**1**

vote

**0**answers

62 views

### Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix:
$C=AB$
where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...

**0**

votes

**0**answers

56 views

### Inverse Laplace transform of matrix exponential

I have the following Laplace-transformed, matrix-valued function:
$$U(s) = e^{As + B},$$
where $A$ and $B$ are diagonalizable, noncommuting (but very close to commuting, if that's useful -- $B$ is ...

**7**

votes

**3**answers

187 views

### Is there a standard name for the following type of linear operator?

Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...

**5**

votes

**5**answers

560 views

### Elementary linear algebra over a (possibly skew) field $K$

I have a number of questions which seem linked to me, about basic (?) linear algebra:
Given a field (possibly skew) $K$, and an superfield $L$, one can do linear matrix algebra with coefficients in ...

**3**

votes

**1**answer

88 views

### Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]

EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore
Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...

**9**

votes

**2**answers

242 views

### Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$

I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...

**1**

vote

**1**answer

60 views

### Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces

Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by
$$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$
$$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$
I wish to find their convex hull, that is a ...