**2**

votes

**1**answer

34 views

### Bounds on Matrix Exponential

Suppose $A$ and $B$ are (non-commuting) hermitian $n\times n$ matrices and $k$ is a large positive number. Suppose we write the product of matrix exponentials as
$e^{kA + B} e^{-kA} = e^{C(k)}$
for ...

**2**

votes

**1**answer

48 views

### Eigenvalues of partial Hankel matrices

I was wondering if there are closed formulas for the singularvalues of a partial Hankel matrix (by partial I mean $\ell<n$)
\begin{align*}
H=
\begin{bmatrix}
c_1 & c_2 & \ldots & c_\...

**3**

votes

**1**answer

129 views

### Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$.
My question is the following:
Does it always exists a ...

**6**

votes

**0**answers

120 views

### Minimizing the operator norm of a sum of matrices

Given an $m\times n$ real matrix $C$.
Let $a_i\in\{-1,1\}$ and $b_i\in\{-1,1\}$. Consider
$$q=\left\|\text{diag}(a_1,\ldots,a_m)C + C\text{diag}(b_1,\ldots,b_n)\right\|$$
where the norm is the ...

**-1**

votes

**0**answers

62 views

### When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?

**0**

votes

**0**answers

34 views

### Is that possible to use stieltjes transform for multiple matrices

I have the matrix calculation with expression
\begin{equation}
\frac{1}{M}tr(\mathbf{WHH}^H\mathbf{W}^H + \mathbf{R}_{nn})^{-1}
\end{equation}
whereas $\mathbf{H} \in \mathbb{C}^{M\times K}$, $\mathbf{...

**5**

votes

**2**answers

142 views

### Lower bound for smallest eigenvalue of symmetric doubly-stochastic Metropolis-Hasting transition matrix

For my master's thesis research, I stumbled upon a question concerning the Metropolis-Hasting transition matrix $W$.
Context $\quad$
Let me start with some context. I consider connected undirected ...

**10**

votes

**3**answers

847 views

### Determinant of a $k \times k$ block matrix

Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...

**1**

vote

**0**answers

45 views

### max min of ratio of quadratic forms

Consider the optimization over two vectors $x$ and $y$
$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$
for two positive definite matrices $A$ and $B$.
This problem can be ...

**0**

votes

**0**answers

34 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, $$\lvert\lambda_1-...

**2**

votes

**1**answer

177 views

### Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com
http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe
Now I ...

**6**

votes

**3**answers

212 views

### Logarithms of matrices in the disk-algebra

It is easy to see that within the disk algebra $A(D)$
$$\Delta(z):= \begin{pmatrix} 1&0\\z&1 \end{pmatrix}\; \begin{pmatrix} 1&1\\0&1 \end{pmatrix}=
\begin{pmatrix} 1&1\\z&1+...

**3**

votes

**1**answer

307 views

### Singular values of the sum of A and A^T

As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}(A+A^{T})$ for a stochastic $A$ (as a function of the singular values of $A$), which is generally ...

**1**

vote

**3**answers

164 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 $M_{n}(\mathbb{C})$ ...

**9**

votes

**2**answers

481 views

### Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
...

**1**

vote

**0**answers

36 views

### Matrix diagonalization after a completely positive transformation

I have a hermitian matrix $A$ which can be diagonalized:
$$A=UDU^+,$$
where U is the unitary matrix and D is the diagonal matrix.
Next, I have a completely positive transformation over it, which is ...

**0**

votes

**0**answers

43 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. ...

**0**

votes

**1**answer

60 views

### Restricted Isometry Property for Discrete Fourier Transform Matrix

I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\...

**3**

votes

**2**answers

286 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 ...

**1**

vote

**0**answers

65 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 ...

**3**

votes

**1**answer

97 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 ...

**1**

vote

**0**answers

51 views

### inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation.
...

**0**

votes

**1**answer

164 views

### $P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and
$P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable.
$Z$ is eigenvalue of $P(Z )$ if $\...

**5**

votes

**1**answer

85 views

### Optimization of a function of a positive definite matrix and its inverse

This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.
Suppose I have two real, positive ...

**19**

votes

**0**answers

573 views

### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...

**4**

votes

**4**answers

927 views

### The derivative of the Cholesky Factor

Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be it's Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...

**2**

votes

**0**answers

66 views

### When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?

It would be soo awesome if you could help!
For $ p \in \mathbb{N}$ consider the following $\mathcal{S(H)}^{p\times p}$-matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p}$ of nuclear operators ...

**7**

votes

**1**answer

78 views

### Add a multiple of $I$ to a matrix to minimize its operator norm

Given $A\in\mathbb{C}^{n\times n}$, what is $s_* = \arg\min \|A-sI\|$?
Here $\|A\|$ is the operator norm, $\|A\|=\rho(A^*A)^{1/2}$, and $I$ is the identity.
The corresponding problem for the ...

**4**

votes

**0**answers

59 views

### Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...

**1**

vote

**0**answers

57 views

### Log convexity for the norm of a vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...

**1**

vote

**0**answers

44 views

### Triangle inequality for nonconvex functions of singular value vector

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...

**0**

votes

**1**answer

53 views

### Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...

**1**

vote

**0**answers

99 views

### Boundary of pseudospectra

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 ...

**-1**

votes

**1**answer

5k views

### Derivative of log determinant and inverse

I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...

**1**

vote

**1**answer

387 views

### sign-flipping inverse

Consider this matrix:
$Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$
Its inverse is entrywise negative (you can check...) and ...

**2**

votes

**0**answers

75 views

### Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?

**1**

vote

**0**answers

38 views

### Limits on a parameter $\alpha$ to get positive definite matrix

Given positive semi-definite $n\times n$ matrices $B_k$, how would I go about getting the limits on $\alpha_k$ such that the expression
\begin{equation}
\mathbb{I}-\sum_{k=1}^{m-1}\alpha_kB_k
\end{...

**5**

votes

**2**answers

221 views

### Induced matrix norm less than one for matrices with spectral radius less than one

Let $A$ be a square matrix with elements in $\mathbb{R}$ or $\mathbb{C}$,
$\rho\left(A\right)$ stands for the spectral radius of $A$, i.e.,
the maximum absolute eigenvalue of $A$; $A^{*}$ is the ...

**4**

votes

**2**answers

163 views

### Non-asympototic version of Gelfand's formula

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...

**0**

votes

**0**answers

39 views

### If $A$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$

If $A \in M_n$ is doubly stochastic and reducible.
Why is $A$ permutation similar
to a matrix of the form $A_1 ⊕ A_2$, in which both $A_1$ and $A_2$ are doubly stochastic?

**1**

vote

**0**answers

75 views

### Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?

Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix
$M = \frac{1}{2}(A + {A^T})$.
Why does $\rho (A) \le {\lambda _{\max }}(M)$?

**16**

votes

**2**answers

1k views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...

**7**

votes

**1**answer

154 views

### approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...

**1**

vote

**0**answers

75 views

### A question on Perron–Frobenius theorem [closed]

Let $A \in M_n$ is nonnegative(all $a_{ij}\ge0$).
Suppose $A$ has a nonnegative eigenvector(all entries$\ge0$ ) with $r ≥ 1$ positive entries and $n − r$ zero entries.
Why is there a permutation ...

**7**

votes

**1**answer

119 views

### Efficient SVD of a matrix without some of the columns

I have a matrix $A \in \mathbb{R}^{p \times q}$ of rank $r$ and its SVD decomposition, i.e,
$$
A = U S V^\top,
$$
where $U \in \mathbb{R}^{p \times r}$ and $V \in \mathbb{R}^{q \times r}$ are ...

**0**

votes

**0**answers

51 views

### On the positive definiteness of RBF kernel with DTW distance

Consider an RBF kernel that is defined as $K(\mathbf{x},\mathbf{y}) = \exp\left(-\dfrac{d^2(\mathbf{x},\mathbf{y})}{2\sigma^2}\right)$, where $d(\mathbf{x},\mathbf{y})$ is usually chosen as the ...

**6**

votes

**2**answers

431 views

### The space of positive definite orthogonal matrices

The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite.
$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal.
and the ...

**5**

votes

**1**answer

120 views

### On proof of the conditionally negative definiteness of a kernel

Let the kernel be $f(\mathbf{x},\mathbf{y}) = \arccos(\mathbf{x}^T \mathbf{y})$, where $\mathbf{x}$ and $\mathbf{y}$ are $\ell_2$ normalized vectors of the same dimensionality, and $\arccos(\cdot): [-...

**4**

votes

**3**answers

168 views

### Fast Upper Triangular Matrix Exponentiation

Let $Q_n$ be a $n\times n$ matrix with $Q_n=\begin{pmatrix} -\lambda_1-\mu_1 & \lambda_1 & 0 & \cdots\\ 0 & -\lambda_2-\mu_2 & \lambda_2 & \cdots\\ \vdots & \vdots & \...