**-5**

votes

**0**answers

23 views

### Finding equivalent matrix combination [on hold]

I have a program I've written that is solving some problems with some matrix-vector math, but I have a feature I want to add and while I've found a work around an analytic solution would be superior. ...

**-3**

votes

**0**answers

30 views

### A question on subordinate matrix norm

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**3**

votes

**0**answers

259 views

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

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 ...

**2**

votes

**0**answers

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

**0**

votes

**0**answers

48 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\ $ ...

**7**

votes

**0**answers

136 views

### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...

**-4**

votes

**0**answers

73 views

### commutable matrices [closed]

I have two 5x5 matrices that don't commute with each other. I want to know, Is there any transformation that when I apply to matrices they change to be commutable and after I exert my mathematical ...

**8**

votes

**1**answer

111 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 $d$) ...

**6**

votes

**0**answers

382 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

**1**answer

71 views

### The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p_1, \dots, p_r$ be $r$ distinct odd primes.
Set $$G = ...

**7**

votes

**1**answer

92 views

### What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$?
Here we denote by $d(G)$ the smallest ...

**2**

votes

**0**answers

48 views

### Integral of a parametrized commutator

I am trying to solve the following integral
$$
\int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2}
$$
where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, ...

**6**

votes

**1**answer

85 views

### “Additive version” of Kronecker product

Let $A$ and $B$ be two square matrices with complex entries.
Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and
$\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$.
Then the Eigenvalues ...

**5**

votes

**1**answer

149 views

### Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements.
Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ?
I am interested in ...

**-1**

votes

**0**answers

49 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, ...

**22**

votes

**3**answers

783 views

### On determinants formed by binomial coefficients

Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers.
Let us form naively a $q \times q$ ...

**15**

votes

**1**answer

460 views

### Is SL(n,Z[x]) generated by transvections?

Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?

**-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 ...

**3**

votes

**0**answers

79 views

### Relating Numerical Range and Perron-frobenius theorem for positive matrices?

Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers
$$W(A)=\{x^HAx\lvert ...

**1**

vote

**1**answer

73 views

### Is this matrix positive semi-definite? [closed]

Consider $K$ vectors $x_1,\dots,x_K$ in $\mathbb{R}^N$. Define the $K\times K$ matrix $A$ whose $(i,j)$ entry is given as $$A_{ij}=\exp(-\frac{||x_i-x_j||^2}{2})$$ Is this matrix Positive ...

**7**

votes

**1**answer

67 views

### Approximation theoretic question about operator norm

Let $\|M\|:=\sup_{u:\|u\|=1}\|Mu\|$ be the operator norm induced by the Euclidean distance.
Suppose $A$ is a $k\times k$ symmetric matrix with $A_{ij}>0$ for all $i,j$ and $\sum_{i,j} A_{ij} = ...

**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 ...

**16**

votes

**0**answers

504 views

### How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...

**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 $a_2$, ...

**7**

votes

**0**answers

105 views

### Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...

**1**

vote

**1**answer

146 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

121 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

**1**answer

153 views

### Is there a bound on the rank of finite index subgroup of SL_3(Z)?

Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?

**1**

vote

**1**answer

74 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

110 views

### A lower bound for orthogonal partial circulant matrices

Let us call an $m$ by $n$ matrix with $m<n$ a partial circulant matrix it is the first $m$ rows of some square circulant matrix.
Consider partial circulant matrices whose elements are either $-1$ ...

**2**

votes

**0**answers

95 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**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 ...

**15**

votes

**2**answers

577 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 ...

**2**

votes

**1**answer

185 views

### Trace of a Product of Finitely Many Matrices with Cosine Entry

Can someone help me prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
2\cos\frac{2j\pi}{n} & -m \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & ...

**2**

votes

**0**answers

32 views

### Generate matrices at the given distance from the initial matrix

I have n-by-n correlation matrix $C_{ij}$, it's symmetric positive definite, and $C_{ii}=1$ and $C_{ij}\le1$ etc.
I want to generate a bunch of matrices $M_{ij}$ which are on the same distance from ...

**0**

votes

**0**answers

46 views

### $l_{\infty}$ norms of matrix perturbations

Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that ...

**2**

votes

**1**answer

138 views

### Algebraically independent matrix invariants

Let $V$ be the space of pairs of $n \times n$ matrices over $\mathbb{C}$ and let $G$ be the space of $n \times n$ permutation matrices which acts on $(A,B) \in V$ by simultaneous conjugation. It is ...

**5**

votes

**0**answers

133 views

### Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$

Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is
\begin{equation}
\Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in ...

**4**

votes

**2**answers

76 views

### Integral roots of circulant matrix

When does the circulant matrix have only integral roots?
For example: all roots of the adjacency matrix of the complete graph $K_n$ are integer, which its adjacency matrix is circulant, but in case ...

**0**

votes

**1**answer

61 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 under such ...

**0**

votes

**0**answers

31 views

### Sparse Matrix Reordering

Matrix reorderings are important for many direct solvers. Sometimes the objective
is to reduce the bandwith or the generated fill in by LU Decomposition.
I am interested in a reordering which reduces ...

**0**

votes

**1**answer

119 views

### Reference request: Strong Connectivity and the Incidence Matrix

Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix?
Details: Consider a digraph $(V, E)$ with vertex set
$$V = ...

**3**

votes

**0**answers

251 views

### Maximizing sum of matrices

Over the last few months, I've been trying to find the solution to a research-related problem I'm having. However, my research is not in mathematics, and my progress toward reaching a solution has ...

**1**

vote

**1**answer

80 views

### Limit of largest eigenvalue [closed]

For positive definite matrix, if we increase the dimension to the infinity, is it true that the largest eigenvalue stays bounded from above?
In other words does the following limit exists:
...

**0**

votes

**1**answer

230 views

### Meaning of $[A,B]$ when $A$, $B$ are self-adjoint

This is just a question about notation, but it got no useful answers on math.stackexchange.
Let $L$ be the Lie algebra of $n\times n$ Hermitian matrices, with Lie bracket $(A,B)\mapsto i(AB-BA)$.
...

**6**

votes

**1**answer

137 views

### Varieties parametrizing skew-symmetric matrices

Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices.
Let $M\in\mathbb{P}(\bigwedge^2V)$, for any ...

**9**

votes

**3**answers

465 views

### Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...

**8**

votes

**1**answer

593 views

### Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...

**11**

votes

**4**answers

439 views

### The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their eigenvalues, what about pairs?

For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of ...

**0**

votes

**1**answer

86 views

### How to write a given rank matrix with some constraints?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.
...