**2**

votes

**0**answers

75 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 ...

**-5**

votes

**0**answers

30 views

### How to convert a row of matrix as ordered pairs in matlab? [on hold]

I have an n*3 matrix.Now i have to plot each row as a point with the elements in first, second,third columns as x,y,z coordinates in 3-d space.can anyone help me do this in matlab?
Thanks in advance

**6**

votes

**2**answers

862 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 ...

**3**

votes

**1**answer

162 views

### Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$

I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...

**2**

votes

**1**answer

179 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 & ...

**0**

votes

**0**answers

63 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 ...

**-5**

votes

**0**answers

53 views

### $A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\} \Rightarrow$A is closed [on hold]

Suppose
${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...

**-5**

votes

**0**answers

45 views

### Let $A = \left\{ {\lambda \in C:{v^*}P(\lambda )v = 0,v \in C,{v^*}v = 1} \right\}$ Why is $A$ always closed? [on hold]

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

**-2**

votes

**0**answers

42 views

### A question in numerical range of matrix polynomial [on hold]

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

**14**

votes

**2**answers

472 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 ...

**-4**

votes

**0**answers

40 views

### Commutator of a matrix as matrix multiplication [closed]

I want to find whether two square matrix A and B are commmutative as a multiplication either A is a contant multiplication of the identity matrix or the matrix B can be expressed as p(A) where p(x) is ...

**-2**

votes

**0**answers

32 views

### Taking a 3d to 2d point [closed]

I'm trying to create a billboard type effect (orienting an object in 1/2 axis' excluding the other) using this code.
...

**2**

votes

**0**answers

30 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

37 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 ...

**6**

votes

**2**answers

656 views

### Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...

**2**

votes

**1**answer

120 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 ...

**-1**

votes

**0**answers

45 views

### Eigenvalues of circulant matrices [closed]

How to prove that the eigenvalue of cycle $C_n=\lambda_r=2 cos(2\pi r/n)$?where $r=0,1,...n-1$, which is proved for the circulant matrix with first row $(v_0=0,v_1=1,v_2=0, ...v_{n-2}=0,v_{n-1}=1)$, ...

**4**

votes

**2**answers

64 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 ...

**5**

votes

**0**answers

118 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 ...

**3**

votes

**1**answer

101 views

### Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...

**7**

votes

**2**answers

1k views

### On positive matrices and their eigenvectors

Let $A$ be an $n\times n$ positive integer-valued matrix, that is every entry of $A$ is a a positive integer. Let $\lambda$ be the Perron-Frobenius eigenvalue and $x = (x_1,...,x_n)^T$ the ...

**0**

votes

**0**answers

24 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 ...

**0**

votes

**0**answers

24 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

88 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 = ...

**12**

votes

**2**answers

424 views

### Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...

**0**

votes

**1**answer

160 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 $(A,B)\mapsto i(AB-BA)$.
...

**1**

vote

**0**answers

95 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 ...

**19**

votes

**6**answers

3k views

### Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n ...

**1**

vote

**1**answer

72 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:
...

**141**

votes

**20**answers

19k views

### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
This question ...

**0**

votes

**1**answer

85 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 $2$.
...

**9**

votes

**3**answers

458 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}
...

**2**

votes

**1**answer

196 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**6**

votes

**1**answer

129 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 ...

**11**

votes

**4**answers

418 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 ...

**8**

votes

**1**answer

493 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 ...

**-1**

votes

**0**answers

47 views

### Spectral radius of principal submatrices for the case of hermitian matrix

A principal submatrix of a matrix $\mathbf{A}\in\mathbb{C}^{N\times N}$ is any submatrix from $\mathbf{A}$ for which the same rows and columns have been eliminated.
Assume $\mathbf{A}$ is hermitian, ...

**4**

votes

**1**answer

177 views

### Hankel matrix commuting with a Jacobi matrix

Assume the semi-infinite Hankel matrix $H$ with entries
$$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$
where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a ...

**-3**

votes

**1**answer

4k views

### Product of Positive Matrices

Is the product of non-negative definite matrices also non-negative definite? If not, let A and B be non-negative definite matrices, is '$\operatorname{tr}(A^T B) \ge0$' ?

**2**

votes

**1**answer

207 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**

vote

**0**answers

364 views

### Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...

**4**

votes

**2**answers

226 views

### Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case
$A = \begin{pmatrix} ...

**3**

votes

**1**answer

142 views

### Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$?
My intention ...

**2**

votes

**1**answer

63 views

### Complexity of sparse matrix-vector multiplication?

I have a vector $\mathbf{x}$ of size $m\cdot n$ of zeros and ones, i.e., $\mathbf{x}\in\{0,1\}^{m\cdot n}$ and a matrix $\mathbf{A}$ of size $\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)$ of ...

**2**

votes

**2**answers

124 views

### What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?

**1**

vote

**0**answers

66 views

### How to calculate $det(X^TX)$ efficiently, update one column of X each time [closed]

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$.
First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...

**0**

votes

**2**answers

338 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

**1**

vote

**0**answers

37 views

### Most general hypotheses to have matrix-exponential as solution of linear matrix-ODE? [closed]

What are the most general hypotheses under which the exponential function $\exp(-B(t))$, with $B(t)$ the indefinite integral of a real Lebesgue-integrable $d\times d$-matrix function $A(t)$, solves ...

**2**

votes

**1**answer

100 views

### Enumerator Polynomials for Linear Anytime Codes

Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...

**5**

votes

**0**answers

54 views

### Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...