Questions tagged [matrix-exponential]

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

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0answers
79 views

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 $...
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74 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\...
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0answers
77 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} ...
11
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0answers
648 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, ...
12
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4answers
712 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 $U_n$ (...
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0answers
40 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(\...
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0answers
70 views

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 &...
1
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1answer
71 views

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}...
6
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1answer
638 views

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 ...
4
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0answers
133 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 ...
3
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1answer
88 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} (...
9
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1answer
310 views

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 ...
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0answers
155 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 ...
3
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2answers
727 views

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 ...
5
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3answers
497 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. ...
1
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0answers
121 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 ...
6
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1answer
805 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 ...
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2answers
4k 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 ...
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1answer
302 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 ...
7
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3answers
2k 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 & \...
2
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0answers
319 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 ...