**0**

votes

**0**answers

35 views

### How the idea of adjoint matrix has been designed? [on hold]

I can understand the adjoint 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 ...

**5**

votes

**1**answer

110 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} & ...

**-3**

votes

**0**answers

44 views

### What are eigenbasis of binary brownian motion? [on hold]

I am seeking for a closed-form expression of eigenbasis for binary brownian motion sign(cumsum(randn(n,1))). Eventually, I need an associate fast transform for this ...

**2**

votes

**0**answers

31 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 $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...

**4**

votes

**0**answers

104 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 = ...

**0**

votes

**0**answers

36 views

### Nonnegative matrix times positive diagonal matrix's spectral bound

Assume A is a matrix. Denote $s(A)$ the spectral bound of A, which means the maximum real part of the eigenvalues and donate $r(A)$ the spectral radius. Now suppose $M$ is nonnegative $n\times n\ $ ...

**-4**

votes

**0**answers

38 views

### Modules theory and Catagories theory [on hold]

What is the main purpose we studied about modules theory and catagories theory (homomorphism)?
what is the function, why there is modules and homomorphism?
i confused when a non mathematics people ask ...

**0**

votes

**0**answers

64 views

### Showing a permutation module is reducible [on hold]

CROSS POST
For a permutation module $V$, which is a permutation module if it has the basis $B = \{v_1,...,v_n\}$ such that the matrix of every $g \in G$ with respect to this basis is a permutation ...

**0**

votes

**0**answers

64 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 ...

**-4**

votes

**0**answers

45 views

### Proof If AB−I is invertible then BA−I invertible [closed]

Proof If AB−I is invertible then BA−I invertible.
I already found an answer online:Here
and Here - the invertibility of I-AB equals to AB-I so I had to multiply (-I)(I-AB)

**8**

votes

**1**answer

108 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 $d$) ...

**14**

votes

**1**answer

447 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 ...

**1**

vote

**0**answers

90 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 ...

**6**

votes

**0**answers

376 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 ...

**1**

vote

**0**answers

68 views

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

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 ...

**-6**

votes

**0**answers

87 views

### How to expand the result of a transpose matrix times the original matrix? [closed]

I am struggling with this one. I am not sure how I can get the second step from the first step. What I get is
...

**6**

votes

**1**answer

220 views

### Geometric meaning of unimodular matrix

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

**-1**

votes

**0**answers

47 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, ...

**-2**

votes

**0**answers

66 views

### How to find composition series for $GL_{2}(\mathbb{Z}/5\mathbb{Z})$ and $GL_{2}(\mathbb{Z}/25\mathbb{Z})$ [migrated]

I'm having trouble trying to find composition series for $GL_{2}(\mathbb{Z}/5\mathbb{Z})$ and $GL_{2}(\mathbb{Z}/25\mathbb{Z})$. I have no idea how to construct this... Someone says that the number ...

**3**

votes

**1**answer

121 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 ...

**0**

votes

**0**answers

39 views

### Build polynomial orthogonal to set of other pre-defined polynomials [migrated]

Basically. the question is simple.
Is there any algorithm so I can build a polynomial, orthogonal for the set of pre-defined polynomials? I need the algorithm(like Gram-Schmidt) that would be ...

**-2**

votes

**0**answers

102 views

### Free abelian subgroups of SL(3,$\mathbb{Z}$) [migrated]

Does SL(3,$\mathbb{Z}$) have any free abelian subgroup of rank > 2?
I want to find 3 $\times$ 3 integer matrices with determinant 1 such that the matrices are commutative, but there exists no other ...

**4**

votes

**0**answers

113 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 ...

**1**

vote

**2**answers

93 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 ...

**2**

votes

**0**answers

56 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 + ...

**2**

votes

**1**answer

111 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

**0**

votes

**0**answers

20 views

### Upper Bound the Linear Combination of Various Matrix Products

Let K and L be symmetric PSD matrices of size N*N, with all entries in [0,1]. Let i be any number in {1...N} and K', L' be two new symmetric PSD matrices, each with only row i and column i different ...

**0**

votes

**0**answers

38 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 ...

**5**

votes

**0**answers

123 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 ...

**-1**

votes

**0**answers

42 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 ...

**8**

votes

**1**answer

229 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 $f: H\to GL_m(\mathbb{Z})$ ...

**0**

votes

**0**answers

28 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 $a_2$, ...

**0**

votes

**1**answer

81 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 ...

**2**

votes

**0**answers

49 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

145 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 $n$
...

**-4**

votes

**1**answer

120 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

110 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

73 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

224 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**

votes

**0**answers

60 views

### writing certain diagonal element as a product of unipotents in the unitary group

In the group $SL_2(\mathbb C)$ I can write the diagonal matrix $diag(t,t^{-1})$ as a product of 6 unipotents as follows: $x_{12}(t)x_{21}(-t^{-1})x_{12}(t) x_{12}(-1)x_{21}(1)x_{12}(-1)$ where ...

**1**

vote

**1**answer

69 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

64 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) = ...

**-2**

votes

**0**answers

130 views

### A question in matrix polynomial

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

**4**

votes

**2**answers

92 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

74 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

64 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 ...

**0**

votes

**0**answers

29 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

156 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

111 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

106 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 ...