Skip to main content
MathOverflow

Questions tagged [matrix-exponential]

Ask Question

The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

45 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
2 votes
1 answer
312 views

For a matrix $M$, let $M^*$ denote the Hermitian conjugate of $M$. Fix integers $d \geq 1$ and $m \geq 1$. Consider the matrix-valued function $f : {\Bbb C}^{d \times m} \to {\Bbb C}^{d \times d}$ be ...
ssss nnnn's user avatar
  • 309
2 votes
0 answers
118 views

Let $U \in U(n)$ be a generic unitary matrix. Since the unitary group $U(n)$ is compact and connected, I know that the exponential map is surjective, i.e. that every $U \in U(n)$ has the form $U = e^{...
William Schober's user avatar
  • 21
2 votes
0 answers
66 views

Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc} A & C\\ C^T & B\\ \end{array}\right)$ ...
Ben94's user avatar
  • 21
2 votes
1 answer
227 views

As in the object, I'm looking at the case where $x \in \mathbb R^d$ is a generic vector, and $AA^\top \in \mathbb R^{d \times d}$ is a p.s.d. matrix. I'm investigating the following inequality $x^\top ...
Simone256's user avatar
  • 69
6 votes
1 answer
348 views

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as $$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$ where ...
rosan98's user avatar
  • 471
1 vote
2 answers
291 views

Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
KatsanikJr's user avatar
  • 21
2 votes
0 answers
201 views

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...
Pavel Gubkin's user avatar
  • 607
8 votes
2 answers
791 views

It is known that an entire function that is nowhere zero must be the exponential of another entire function. Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
Kanghun Kim's user avatar
  • 296
5 votes
2 answers
610 views

Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that $ A=\exp L $...
Bazin's user avatar
  • 16.7k
4 votes
0 answers
424 views

$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
NIshant Rathee's user avatar
  • 41
5 votes
0 answers
188 views

$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...
Marco's user avatar
  • 325
3 votes
1 answer
549 views

I have non-negative $d\times d$ matrices $A$, $B$ and need a tractable way to compute the sum of all entries of $\exp(-t(A-B))$ where $A$ is diagonal and $B$ symmetric rank-$1$. IE $$f(t)=\langle\exp(-...
Yaroslav Bulatov's user avatar
  • 2,394
1 vote
0 answers
202 views

Let $p:E \to B$ be a smooth vector bundle of rank $n$ over a manifold $B$ and we identify $B$ with the image of the corresponding zero section. For $b\in B$ denote by $E_b = p^{-1}(b)$ the fiber over $...
Sergiy Maksymenko's user avatar
  • 859
2 votes
1 answer
205 views

Consider the following exponential of matrices $\exp(X+\delta Y)$, where $\delta$ is a smaller number, and $X,Y$ are non-commuting matrices. I am interested in expanding it in such a way that $$ \exp(...
fagd's user avatar
  • 163
3 votes
1 answer
563 views

Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
Kanghun Kim's user avatar
  • 296
1 vote
1 answer
692 views

Let $A$ be an invertible, symmetric and tridiagonal matrix of size $n \times n$. Assume that $A_{i,i}=a \neq 0$ for $i=1\dotsc n$ and all the elements in the sub- and super-diagonal of $A$ are $b \neq ...
Mirar's user avatar
  • 370
5 votes
1 answer
213 views

Let $X$ be an $n \times d$ random matrix with iid entries from $N(0, 1/d)$. Let $S:=X^\top X/n$, a $d \times d$ Wishart matrix and let $T = e^{S} := \sum_{k=0}^\infty \dfrac{S^k}{k!}$ be its ...
dohmatob's user avatar
  • 7,043
4 votes
0 answers
210 views

This question is crossposted from Math Stackexchange here. I crosspost without much edits as I think this is the best way to phrase the question and because I received no feedback on the original post ...
Afham's user avatar
  • 41
4 votes
0 answers
388 views

Originally posted on Math SE but didn't get any responses. Thus, I thought I would ask here with some more details. I have a matrix originating from Master Equation for birth death process on semi ...
plambda's user avatar
  • 41
1 vote
0 answers
86 views

(I have moved this question from Stackexchange). Given the operators $\{A_j\}$ in a Banach algebra and a positive integer $p$, let \begin{equation} g=\exp\left(\frac{1}{n}\sum_{j=1}^p A_j\right)\quad\...
user96233's user avatar
  • 111
3 votes
0 answers
288 views

Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...
user149918's user avatar
  • 121
2 votes
0 answers
151 views

Let $X$ be a complex Banach space and $A:X \to X$ a compact operator. It spectrum is the set $\sigma(A)=\lbrace \lambda \in \mathbb{C}, \ A-\lambda I \text{ is not invertible}\rbrace$. Let $L=\sup\...
JULIAN EPSTEIN's user avatar
  • 21
2 votes
1 answer
422 views

I want to find the matrix $\hat{H}_d$ which, when exponentiated, leads to a d-dimensional cyclic permutation transformation matrix. I have solutions for d=2: $$ \hat{U}_2 =\left( \begin{matrix} ...
Mario Krenn's user avatar
  • 205
11 votes
0 answers
775 views

As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ My question is, ...
user avatar
14 votes
4 answers
2k views

Let $k$ be a field (of possibly positive characteristic), let $U_n$ denote the space of all $n \times n$ unipotent upper triangular matrices over $k$, and let $G$ be an algebraic subgroup of $U_n$ (...
Mike Crumley's user avatar
  • 521
1 vote
0 answers
62 views

For simplicity, let us work on an example. Regard $GL_r(\mathbb C)$ as a Lie group with associated Lie algebra $M_r(\mathbb C)$, then there exists a canonical so called exponential map: $$\exp: M_r(\...
Longma's user avatar
  • 169
1 vote
0 answers
301 views

Context I'm trying to numerically solve the following differential equation: $\frac{\mathrm{d} u}{\mathrm{d} t} = -Au + f$, where $u$ and $f$ are vectors, and $A$ is an $N \times N$ matrix, with $N &...
MPA's user avatar
  • 119
1 vote
1 answer
189 views

The problem statement is the following: $$U=\exp\{iV\}$$ where $U$ is a unitary unimodular matrix of the following form: $$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
john melon's user avatar
  • 113
11 votes
2 answers
524 views

This question was originally posted on Mathematics SE, and I'm cross posting it here. Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 &...
Yly's user avatar
  • 996
8 votes
1 answer
2k views

Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements? Motivation: I'm trying to find the first passage time ...
stochastic's user avatar
  • 371
4 votes
0 answers
160 views

Assume I am given an $n \times n$ matrix $A$ with real or complex coefficients. Its matrix exponential is denoted by $\exp(A)$ and is calculated as usual. Assume further that I want to rescale the ...
tobias's user avatar
  • 759
3 votes
1 answer
163 views

Let $A$ be a (finite) Hurwitz matrix. In this related question of mine, (see also https://en.wikipedia.org/wiki/Lyapunov_equation) it is shown that $$ \int_0^\infty \sum_{j,k} (e^{At})_{ij} Q_{jk} (...
N. Gast's user avatar
  • 562
9 votes
1 answer
649 views

The Baker-Campbell-Hausdorff problem is to obtain $\log(e^{X}e^{Y})$ where $X,Y$ are appropriate operators. The Dynkin series $$\log(e^{tX}e^{tY})=t(X+Y)+\frac{t^2}{2}[X,Y]+o(t^3)$$ gives an expansion ...
Semiclassical's user avatar
  • 319
0 votes
0 answers
328 views

Let $A$ and $B$ be commuting $n\times n$ positive definite complex matrices and let $C$,$D$ be other complex matrices. I wish to think of $C,D$ as small and $A,B$ as any size. Suppose we are looking ...
JRoss's user avatar
  • 250
3 votes
2 answers
1k views

There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as: All eigenvalues of $A$ have negative ...
Rubi Shnol's user avatar
  • 801
5 votes
3 answers
904 views

Note: I posted my question on math.stackexchange but got no answer. That is why I am asking it here. Let $A$ be a $n\times n$ square matrix such that the real part of all eigenvalues are negative. ...
N. Gast's user avatar
  • 562
1 vote
0 answers
202 views

Let $A$ be a stochastic matrix, $q\in (0,1)$. How to bound $n$ such that $$q^n A^n e^A \leq e^A$$ Note that here $e^A$ is the matrix exponential, and $\leq$ is taken entrywise. To be clear, what I ...
maomao's user avatar
  • 502
6 votes
1 answer
1k views

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 ...
JRoss's user avatar
  • 250
14 votes
2 answers
9k views

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 ...
FaCoffee's user avatar
  • 261
0 votes
1 answer
556 views

I am looking for identities that may help with numerical computation of the matrix exponential ${\rm exp}(A)$ where ${\rm tr}(A)=0$. I am already aware of general-purpose algorithms for computing the ...
Alex Flint's user avatar
  • 561
9 votes
3 answers
4k views

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 & \...
Alex R.'s user avatar
  • 5,012
62 votes
7 answers
10k views

Given matrices $$A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr)$$ where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following? $$\det \big( I + e^...
Lei Wang's user avatar
  • 835
7 votes
2 answers
888 views

For three symmetric positive semidefinite matrices $A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases: $$ \operatorname{tr} \left( A e^{B+C} \right) \leq \...
nikka's user avatar
  • 385
2 votes
0 answers
555 views

I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here. What ...
Winton's user avatar
  • 21
16 votes
4 answers
3k views

In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
Suvrit's user avatar
  • 28.7k