**0**

votes

**0**answers

9 views

### How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparce rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...

**0**

votes

**0**answers

23 views

### Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, ...

**0**

votes

**1**answer

47 views

### Matrix norm inequality for C*-Algebras [on hold]

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, ...

**0**

votes

**0**answers

26 views

### Lower bound of general bilinear form [on hold]

Suppose I have a bilinear form $X^TAY$ where $X \in R^n, Y \in R^m$ and $Α \in R^{n \times m}$. All elements of $A$ are bounded, that is $\exists \bar a_{ij}>0:|a_{ij}|\le \bar a_{ij}, \forall ...

**1**

vote

**0**answers

38 views

### Matrix transformation [on hold]

I want to show that
$(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis
with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$
is equivalent to $M_\gamma$ has no purely imaginary ...

**0**

votes

**0**answers

71 views

### Understanding the significance of the values of an adjacency matrix [closed]

I am new to network/graph theory and I am trying to understand a few things. One of these is the significance of the i,j entry of an Adjacency Matrix. In an ...

**5**

votes

**2**answers

762 views

### What is the Time Complexity of the Matrix Exponential?

While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...

**1**

vote

**0**answers

61 views

### Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...

**5**

votes

**1**answer

175 views

### A sufficient condition (or not) for positive semidefiniteness of a matrix?

Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite.
I do not think the ...

**4**

votes

**1**answer

89 views

### Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge ...

**1**

vote

**0**answers

62 views

### coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant ...

**2**

votes

**1**answer

155 views

### The class of $(-1,0,1)$-matrix with all row sums and column sums equalling to $0$

Let $n$ be an even positive integer and $W_n$ be the class of all $n\times n$ matrices with entries from the set $\{-1,0,1\}$ satisfying all row sums and column sums are equal to $0$.
For any $M\in ...

**5**

votes

**2**answers

276 views

### Explicit solution to a Rayleigh quotient equation

For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
Main equation:
$$
...

**4**

votes

**1**answer

71 views

### Perturbations on the pseudoinverse of a matrix

Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation
$$
A_p = A + \Delta
$$
is there a way to represent
$$
(A_p)^{\star}= (A)^{\star} + f(\Delta)
$$
where $(A_p)^{\star}$ ...

**11**

votes

**3**answers

221 views

### Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

I've read that if $M_1, \dots, M_n$ are matrices in $\mathrm{SL}(2, \mathbb{Z})$ whose product is the identity, and each is conjugate to the shear
$$ \begin{pmatrix} 1 & 1 \\ 0 & ...

**0**

votes

**1**answer

80 views

### Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...

**3**

votes

**0**answers

84 views

### Finding nearest Toeplitz matrix to a given matrix

For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz ...

**0**

votes

**2**answers

52 views

### Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix.
I've seen a similar ...

**4**

votes

**1**answer

82 views

### I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?

This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated.
Question
I have a very ...

**3**

votes

**1**answer

125 views

### Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...

**1**

vote

**0**answers

11 views

### non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...

**9**

votes

**0**answers

196 views

### Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even.
Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...

**1**

vote

**1**answer

81 views

### Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that ...

**-2**

votes

**1**answer

93 views

### Eigenvalues of cyclic tridiagonal matrix [closed]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that this matrix ...

**2**

votes

**1**answer

78 views

### Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...

**1**

vote

**1**answer

41 views

### range of singular values of sub-matrices

Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and ...

**2**

votes

**0**answers

102 views

### Can the matrix exponential be equal to the elementwise exponential [closed]

Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential ...

**1**

vote

**3**answers

163 views

### The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra

Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X)
\otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...

**1**

vote

**1**answer

110 views

### Modified interlacing of eigenvalues

Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A ...

**2**

votes

**2**answers

931 views

### Encoding vectors of size $n$ in matrices which less than $2n$ rows [closed]

I have a set of vectors and each has $n$ nonnegative entries.
Moreover, each entry of a vector has a quality: (1) or (2). It makes $2^n$ different possible patterns.
For example, let's take two ...

**1**

vote

**0**answers

38 views

### Explanation for a spectral measure [closed]

Could someone help me, please, to understand in term of entries of a Matrix $M=(m_{i,j})_{i,j\in\{1,n\}^2}$ the following measure :
$$ \frac1{n} \sum_{i=1}^n \langle v_i,e_j \rangle ...

**3**

votes

**0**answers

161 views

### Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix
$$A = \left(\begin{array}{}
1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots ...

**0**

votes

**1**answer

23 views

### Identifying winner of tournament (A,B testings) with none binary test result [closed]

I'm trying to setup a tournament based on votes. Let say user vote for products A, B and C. Each user is presented all possible combinations of products in random order and he picks his preferred ...

**3**

votes

**0**answers

88 views

### How do I ensure that my matrix is positive definite? [closed]

I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form
\begin{equation}
M=\begin{pmatrix}
\Sigma&P'\\
P&\Sigma
\end{pmatrix}
\end{equation}
where $\Sigma$ is a ...

**0**

votes

**0**answers

42 views

### How to get Wiener-Hopf decomposition of this matrix function

I met a matrix function $M(z)$ that I need to get its Wiener-Hopf decomposition that $M(z) = M_+(z) M_-(z)$ with $M_+(z)$ and $M_-(z)$ being analytic inside and outside the unit circle respectively. ...

**1**

vote

**1**answer

90 views

### Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists.
Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory).
Preferably the ...

**1**

vote

**0**answers

52 views

### The state-transition-matrix of a physical system,

Here's a simple but potential research problem that I am learning about.
Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...

**1**

vote

**1**answer

80 views

### Largest element in inverse of a positive definite symmetric matrix [closed]

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...

**6**

votes

**1**answer

135 views

### Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$).
Can we prove that the following problems are equivalent:
$$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...

**2**

votes

**0**answers

62 views

### SVD when only off-diagonal terms are known

I have a real matrix $A \in \mathbb{R}^{n\times n}$ such that:
$A$ is symmetric
All the off-diagonal terms are known and positive
Has rank $k<n$
Unfortunately I don't know the values of the ...

**19**

votes

**1**answer

1k views

### A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...

**1**

vote

**1**answer

26 views

### how to force least square solution matrix to be diagonal [closed]

Let's say I have the following equation
$$AX=B$$
where $A$ is a $8\times 3$ matrix (known), $X$ is a $3\times3$ "diagonal" matrix which represents the coefficients (unknown) and $B$ is a ...

**1**

vote

**3**answers

119 views

### Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...

**3**

votes

**1**answer

88 views

### Wiener-Hopf factorization of matrices

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are ...

**2**

votes

**1**answer

91 views

### Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form
$$
|D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)}
$$
providing all values in $(\cdot)>0$.
...

**1**

vote

**1**answer

75 views

### If the two smallest eigenvalues of the Laplacian matrix of a network are equal to zero, then does it mean that the network is not connected? [closed]

What does it mean if the two smallest eigenvalues of the Laplacian matrix of a graph are equal to zero?

**-1**

votes

**1**answer

112 views

### Representations of the $3\times 3$ Heisenberg group [closed]

I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations.
Following this article given a symplectic bilinear form $\langle, ...

**2**

votes

**0**answers

35 views

### Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications?
For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature.
In some of ...

**3**

votes

**0**answers

98 views

### An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...

**3**

votes

**2**answers

276 views

### Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries

This is a more carefully worded version of this question, here tailored to professional mathematicians.
Consider a matrix ${\bf A}\in{\bf M}_{n\times n}({\mathbb R})$ with possibly positive, negative ...