Questions tagged [matrix-exponential]

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

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Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well

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
5 votes
2 answers
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Logarithm of a bounded operator

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
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What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?

$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
NIshant Rathee's user avatar
5 votes
0 answers
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Is every linear Lie group of bounded geometry?

$\newcommand\norm[1]{\lVert#1\rVert}$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
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3 votes
1 answer
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Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?

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
1 vote
0 answers
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Fiberwise exponential map for vector bundle automorphisms

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
2 votes
1 answer
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Reorganizing the terms in the Baker–Campbell–Hausdorff formula (or Zassenhaus formula) for $\exp(X+\delta Y)$ for small $\delta$

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
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3 votes
1 answer
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Is the set of real matrices with at least one real logarithm closed under multiplication?

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
1 vote
1 answer
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Kronecker product: Is it possible to simplify this product $e^{-A} \otimes e^{A}$ where $A$ is an invertible and symmetric matrix [closed]

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
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Limiting value of expectation of trace of exponential of Wishart matrix

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
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4 votes
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Matrix logarithm of unitary factor from polar decomposition of product of positive definite matrices

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
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Exponential of infinite dimensional matrix

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
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1 vote
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Inequality of exponentials of Banach operators

(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
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Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators

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
2 votes
0 answers
98 views

Growth bounds for the exponential of an operator

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
1 vote
1 answer
292 views

Matrix logarithm for d-dimensional cyclic permutation matrix

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
11 votes
0 answers
718 views

What properties characterize the function $L(x) = x+\exp(x) \log(x)$?

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
13 votes
4 answers
2k views

exponential/logarithm for unipotent algebraic groups

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
1 vote
0 answers
45 views

Possible compatible ""structure" on the Lie algebra associated with a Lie group with respect to group structure

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
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Matrix exponential with particular structure

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
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1 answer
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Conditions to obtain a real logarithm of a unitary unimodular complex matrix?

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
7 votes
1 answer
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Matrix elements of exponential of tridiagonal matrices

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
4 votes
0 answers
139 views

Is there a nice way to express a matrix exponential when rows are proportionally scaled?

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
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3 votes
1 answer
111 views

Higher order Lyapunov equation

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
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9 votes
1 answer
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Behavior of a Baker-Campbell-Hausdorff problem at infinity

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
1 vote
0 answers
240 views

Product of exponentials of matrices

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
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3 votes
2 answers
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Properties of matrix exponential without using Jordan normal forms

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
5 votes
3 answers
810 views

Integral of the entrywise square of the exponential of a matrix

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
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1 vote
0 answers
164 views

Matrix exponential bounds

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
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6 votes
1 answer
1k 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 ...
JRoss's user avatar
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12 votes
2 answers
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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 ...
FaCoffee's user avatar
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0 votes
1 answer
444 views

Efficient computation of matrix exponential of trace zero matrix [closed]

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
9 votes
3 answers
3k 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 & \...
Alex R.'s user avatar
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63 votes
7 answers
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How to prove this determinant is positive?

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
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7 votes
2 answers
718 views

An extension of the Golden-Thompson inequality

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
  • 375
2 votes
0 answers
427 views

Random variable matrix exponential

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
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