**5**

votes

**1**answer

187 views

### Plethysm of $S^3(S^2V)$ as $\mathfrak{sl}_3(\mathbb{C})$-module

I have asked this question in MSE before, but have not got any answer. So here I am asking it again with some more detail.
I believe that the following sequence of ...

**2**

votes

**0**answers

27 views

### Constrained absolute orientation of 3D point sets

Let us assume we have two 3D point sets, $P=\{p_i\}$ and $Q=\{q_i\}$, and that we need to recover the transformation that takes $P$ as close to $Q$ as possible. In particular, I am interested in ...

**2**

votes

**1**answer

64 views

### an inequality about kronecker product with eigenvalues question

Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of ...

**7**

votes

**2**answers

219 views

### Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?

If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...

**5**

votes

**0**answers

322 views

### How the idea of adjugate matrix has been designed? [closed]

I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...

**6**

votes

**1**answer

204 views

### Is there a way to simplify the following trace expression?

I'd like to simplify the following expression:
$$\text{tr}\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}_{M\times M} & \mathbf{0}_{M\times N} \\ \mathbf{0}_{N\times M} & ...

**2**

votes

**0**answers

47 views

### Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...

**5**

votes

**0**answers

170 views

### Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...

**1**

vote

**0**answers

67 views

### Eigenvalues of a partitioned self-adjoint matrix

This is a repost of the same question on MSE (with no reply/comment):
http://math.stackexchange.com/questions/1454314/eigenvalues-of-a-partitioned-self-adjoint-matrix
I would be grateful just for a ...

**9**

votes

**1**answer

118 views

### a generalization of gamma matrices

Is it possible to find matrix solutions to the following :
$$\left(\sum_1^m M_k x_k\right)^n=\left(\sum_1^m x_k^n\right)I_d$$
where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) ...

**15**

votes

**1**answer

499 views

### Exponentiation of vector spaces?

This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...

**2**

votes

**0**answers

105 views

### A question on vector space over finite field

Let $\mathbb{F}_{2^\sigma}$ be a finite field of $\sigma$-bit elements, and use $\mathbb{F}_{2^\sigma}^{\ell}$ to denote an $\ell$-dimensional vector space over $\mathbb{F}_{2^\sigma}$. Let $V$ be a ...

**7**

votes

**0**answers

397 views

### Name for an operation on matrices?

Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with ...

**2**

votes

**0**answers

77 views

### What does similar eigenvectors and eigenvalues of two matrices really mean? [closed]

Empirically I've noticed that diagonally dominant matrix G and it's diagonal version D (diagonal elements of G on the diagonal and all other elements are set to zero) produce similar eigenvalues and ...

**8**

votes

**1**answer

241 views

### Geometric meaning of unimodular matrix

Rotations are given by unitary matrices.
What is the geometric meaning of unimodular matrices that are not unitary?

**0**

votes

**0**answers

51 views

### A question on boundary set

Suppose:
${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$
${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, ...

**4**

votes

**1**answer

128 views

### Centralizer of hermitian matrices with zero trace

In Quantum Physics one often has to deal with commutators.
Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero!
One can easily relate it to ...

**5**

votes

**0**answers

125 views

### Relative invariants of $P\oplus P^*$

Let $P$ be a $\mathrm{GL}(V)$-module, and assume that the decomposition of $P$ into irreducible submodules is known. By a relative invariant of the module $P\oplus P^*$, I mean a homogeneous nonzero ...

**2**

votes

**2**answers

160 views

### SVD vs Fourier analysis for data.

Fourier analysis is useful for analysis in the frequency domain. SVD on the other hand is useful for analysis of data, and expressing noise in the data. I have a problem that needs extensive data ...

**3**

votes

**0**answers

62 views

### A system of non-linear equations that is decomposable as a product — uniqueness of solution?

I have a system of non-linear equations
$ a_1=f_0 g_1$
$a_2=f_1 g_1 + f_0 g_2$
$a_3=f_2 g_1 + f_6 g_2 + f_0 g_3 $
$a_4=f_3 g_1 + f_7 g_2 + f_6 g_3 + f_0 g_4 $
$a_5=f_4 g_1 + f_8 g_2 + f_7 g_3 + ...

**3**

votes

**1**answer

113 views

### Orbit of $SO_r$ in $SL_r$

Take the action of the special orthogonal group on the speciale linear group by left multiplication, over $\mathbb C$, how could one identify the quotient space?
Thanks

**1**

vote

**0**answers

62 views

### On triangular Toeplitz matrices

Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist ...

**6**

votes

**0**answers

131 views

### Finding $U,V$ in Thompson's Formula

Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that:
$e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$
Given $a,b \in \mathfrak{su}(4)$ defined by:
$a=J_x ...

**0**

votes

**0**answers

45 views

### Finding spanning vector sets

Let $V$ be the set of all vectors over the non-negative integers. For any two subsets $S$ and $T$ of $V$, define $S + T$ to include:
All vectors in $S$
All vectors in $T$
All vectors that can be ...

**9**

votes

**1**answer

237 views

### Is the free product $\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}$ linear over $\mathbb{Z}$?

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I want to know if $H$ is a ($\mathbb{Z}$)linear group that is to say is there an injective homomorphism $f: H\to GL_m(\mathbb{Z})$ ...

**1**

vote

**0**answers

36 views

### Find Moment condition for generalized method of moments

Consider a scalar system with 2K outputs and K+2 unknowns
$y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$.
The variables $n_{k,\ell}$ are zero mean noise variables.
To estimate $a_1$ and $a_2$, ...

**1**

vote

**1**answer

90 views

### Estimate for differential of inverse map

Let $f: M \to N$ be a diffeomorphism between two riemannian Manifolds. Suppose there exist constants $0 < c \leq C$ such that for all $p \in M$, we have $c \leq |df_p| \leq C$. Here, $df$ denotes ...

**0**

votes

**0**answers

61 views

### construction of grassmannian manifolds as collection of subspaces of Euclidean space

The grassmannian $G_k(\mathbb{R}^n)$ is the collection of all $k$-dimensional linear subspaces of $\mathbb{R}^n$ equipped with the quotient topology. The cohomology ring of $G_k(\mathbb{R}^n)$ has ...

**1**

vote

**1**answer

149 views

### characterize certain type of matrices

I am trying to characterize matrices with a certain property :
Define $U$ as an $n \times n$ matrix (over C or R; you can also assume
that it is unitary or orthogonal if it helps). Now take $n$
...

**-3**

votes

**1**answer

139 views

### Eigenvalues of real symmetric matrix [closed]

Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of
$A < 2 (n - 1).$

**3**

votes

**2**answers

114 views

### Level sets on $SU(4)$

Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$?
Can they be written only in terms of abstract linear maps, not ...

**4**

votes

**1**answer

78 views

### testing singularity of integer matrices

I am looking for the best upper bounds on the bit complexity for testing the singularity of an integer $n\times n$ matrix, where each integer is represented with $k$ bits.
I know the fast method for ...

**4**

votes

**2**answers

242 views

### Riemannian metric of hyperbolic plane

I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector ...

**1**

vote

**1**answer

111 views

### Upper bound for sum of absolute values of eigenvalues of Hermitian matrix

Given a hermitian, but not necessarily positive, sparse matrix $C = (c_{ij}) \in \mathbb{C}^{n \times n}$ and $n \ggg 1$ ($n \approx 2^{100}$) with eigenvalues $\lambda_1 \le \lambda_2 \le \dots \le ...

**4**

votes

**0**answers

66 views

### Basin of Attraction

I have a function $F$ which is defined as follows:
$$
F(x) = \sum_{i=1}^N f(z_i^T x)
$$
where ${z_i}$ are known $m \times 1$ vectors, $x$ is an $m \times 1$ vector, and for $t\in \mathbb{R}$, $f(t) = ...

**5**

votes

**2**answers

108 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 ...

**3**

votes

**1**answer

75 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: ...

**2**

votes

**0**answers

66 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 ...

**5**

votes

**1**answer

299 views

### vector spaces with uncountable dimension and a nice basis

Unlike the reals considered as a vector space over the rationals, I know of a number of nice examples of vector spaces with uncountable dimension that have a nice basis.
For example, the space of ...

**0**

votes

**0**answers

30 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 ...

**4**

votes

**0**answers

160 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 $n$ ...

**0**

votes

**2**answers

123 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

**1**answer

110 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 ...

**3**

votes

**1**answer

184 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 \, ...

**4**

votes

**0**answers

99 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 ...

**0**

votes

**0**answers

81 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 ...

**0**

votes

**0**answers

52 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 ...

**5**

votes

**0**answers

109 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 ...

**15**

votes

**2**answers

594 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 ...

**6**

votes

**0**answers

157 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 ...