Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrix-exponential]

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

0
votes
2answers
45 views

Computing the matrix exponential for a Jordan matrix

How can I compute $e^{At}$ where $A = J_{3}(5)$? That is, $$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix} $$ Using this, how can I write down a basis ...
0
votes
1answer
34 views

Deducing the additive Jordan decomposition

In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$. Is that true the ...
1
vote
1answer
81 views

Is every diagonalizable matrix a matrix exponential?

I know it is true in $\mbox{SL}_2(\Bbb C)$ and I think it is true in $M_n(\Bbb C)$ because if $M=PDP^{-1}$, we might be able to write D as $\exp(E)$ for some $E\in M_n(\Bbb C)$ as the exponential is ...
8
votes
0answers
249 views

Complexification of compact connected Lie groups: do these curves have the same tangent vector?

I'm trying to understand the complexification of Lie groups from page $207$ here and I'm having trouble with a computation. Assume $A, B$ are hermitian metrices, and $k$ is a unitary matrix. I want ...
0
votes
0answers
16 views

Expression involving inverse of block matrices and matrix exponentials

I'm struggling to simplify $B$ which is given by $$B=\left(A^{-1}\right)^TS\left(A^{-1}\right)$$ with S a symmetric matrix of size $2m \times 2m$ and A a matrix given by $$A=\left[\begin{matrix} Ve^{\...
3
votes
2answers
33 views

Integral involving matrix exponential

Is there any way to simplify the integral $$ I = \int_{t_1}^{t_2}e^{\Lambda t} A e^{\Lambda t}\,dt $$ knowing that A is symmetric and Λ is a diagonal matrix?
2
votes
1answer
55 views

Are the 1-parameters subgroups of $SO(3)$ closed?

I'm trying to solve the following question Question: Prove that all $1$-parameters subgroup of $SO(3)$ are closed. Does this statement holds for $SO(n),$ $n>3$? Some comments The $1$-...
2
votes
0answers
26 views

Equation with Exponential matrix

I have the following equation: \begin{gather} (e^{At})^T \cdot P \cdot e^{At} - P = - I \end{gather} where im trying to find the matrix $P$. Is there any name to this equation or some simple ...
2
votes
1answer
42 views

Exponential of Hermitian operator

Let $H$ be a Hermitian matrix with operator norm $||H|| \leq 1$. I am trying to show that for each $\varepsilon > 0$ I can find a $\delta$ such that $$\left|\left|e^{iHt}-\sum_{k=0}^{\delta(t + \...
1
vote
0answers
41 views

Exponential matrix estimate

$B=C^{-1}AC=\left[ \begin{array}{cc} P & 0\\ 0 & Q \end{array} \right]$ consider the system y'=By+G(y). Let $U(t)=\left[ \begin{array}{cc} ...
0
votes
1answer
17 views

Partial derivative of composite 3D rotation in exponential coordinates

I have the exponential coordinate/angle-axis vector $\mathbf{w} \in \mathbb{R}^3$ which is composed of two other angle-axis vectors $\mathbf{w}_0$ and $\mathbf{w}_1$: $\mathbf{w} = Log(Exp(\mathbf{w}...
2
votes
0answers
43 views

Limit of log of norm of exponential of Hamiltonian Matrix equals maximal eigenvalue

Let $A$ be a $2n \times 2n$ Hamiltonian matrix (i.e. $JA$ is symmetric with $J=\begin{pmatrix} 0 & I_n \\ -I_n & 0\\ \end{pmatrix}$). Is it true that $$\lim_{t\to \infty}\frac{1}{t} \log \...
1
vote
1answer
27 views

An operator exponential/commutator question

There is "an important lemma" related to the Baker-Campbell-Haussdorff theorem which says that $$ e^XYe^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]]+\ldots $$ Clearly if $[X,Y]=0$ we get (noting that $e^...
1
vote
2answers
47 views

Derivative of matrix exponential [closed]

What is the derivative of $e^{(x-y)Q}$ with respect to $y$, where $x$ and $y$ are scalars and $Q$ is a transition rate matrix?
0
votes
1answer
25 views

Exponential of a Jordan Block using Cayley-Hamilton Theorem

For the sake of simplicity, I will only consider $2\times2$ matrices. The Cayley-Hamilton theorem allows us to conclude that $$e^{At} = \alpha_0I + \alpha_1 A$$ where $\alpha_0$ and $\alpha_1$ can ...
4
votes
1answer
63 views

If A is a real non-singular square matrix, then there exists a real matrix $B$ such that $e^B$ $=$ $A^2$

If we consider the matrix exponential map on $M_n(\mathbb R)$, then what will be the image set of the exponential map? I have seen this. From there I can say that $exp(M_n(\mathbb C))=GL_n(\mathbb C)$...
0
votes
0answers
44 views

Is $f(t) = e^{X(t)}$ continuous when matrix $X(t)$ is continuous?

I think $f(t) = e^{X(t)}$ is continuous over $t \in \mathbb{R}$ when the complex matrix valued mapping $t\mapsto X(t)$ is continuous - regardless of whether $X$ is infinite or finite dimension. I ...
0
votes
1answer
62 views

Why is the Kronecker sum defined for square matrices?

Background From Wikipedia, if A is an $m\times n$ matrix and B is a $p\times q$ matrix, the the Kronecker product $\mathbf A\otimes \mathbf B$ is the $mn\times nq$ block matrix $$\mathbf A\otimes \...
1
vote
1answer
76 views

Is commutation preserved for $e^A$?

Let $A, B \in \mathbb C^{n \times n}$. Suppose $A$ commutes with $B$. Does $e^A$ necessarily commute with $B$? If that is the case, consider $$S = A + B \exp(-S)$$ where $S$ is an $n \times n$ matrix....
1
vote
1answer
35 views

Hamiltonian system and exponential map — backward

Consider a linear ODE $$ \dot x(t) = A\,x(t). $$ A solution is $x(t)=\exp(t\,A)$ where $\exp$ is defined by $$ \exp(A) = \sum_{n=1}^n\frac{A^n}{n!}. $$ Consider a Hamiltonian system $$ \dot x(t) = \...
0
votes
0answers
47 views

Is the transpose of a matrix exponential with itself positive semi-definite under certain conditions

Given is a matrix $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with linearly independent eigenvectors $\boldsymbol{u}_1,...,\boldsymbol{u}_n$ and corresponding eigenvalues $\lambda_1,...,\lambda_n$. ...
0
votes
1answer
21 views

Is there a proof for the Determinant of the Log-Euclidean-Tensor-Interpolation yielding the geometric mean of the Determinants of its sampling points?

Given a set of $i \in \mathbb{N}$ symmetric-positive-definite-tensors $\boldsymbol{A}_i$ and the corresponding weights $w_i$ with $w_i>0$ and $\sum_i w_i = 1$. The Log-Eucledian-Mean is defined as ...
0
votes
0answers
23 views

Exponential map of Lie brakets

Let $G$ be a compact matrix Lie group and $(\mathfrak{g},[,])$ its Lie algebra. For $x$, $x'$ $\in$ $\mathfrak{g}$, $[x,x']=xx'-x'x$. My question is if there exists a relation between exp$(t[x,x'])$ ...
1
vote
1answer
32 views

Exponential of a Matrix- Repeated AND Complex Eigenvalues

I am seeking a general solution to the initial value problem x' = Ax, x(0) = x_0 that can be written out to include both the eigenvalues and eigenvectors. To cover the case of repeated eigenvalues, ...
0
votes
0answers
19 views

Computing the element-wise logarithm of a matrix exponential more efficiently?

Is there any known way to compute the element-wise logarithm of a matrix exponential more efficiently? Motivation: I am trying to an optimization problem (basically finding a specific Markov ...
0
votes
1answer
23 views

How to extract the screw axis vector and the angle from the exponential coordinates?

Given the 6-dimensional vector of the exponential coordinates of the homogeneous transformation: $S\theta$, where $S$ is the screw axis consisting of the pair $(\omega, v)$ and both of them are $3$ ...
0
votes
0answers
17 views

Confusion concerning solution to linear autonomous system

I have recently learned that a solution $\mathbf{x(t)}$ to a $2 \times 2$ system $$\mathbf{x'} = \mathbf{Ax} $$ with $\mathbf{x_0} = c_1\mathbf{u_1} + c_2\mathbf{u_2}$, where $\mathbf{u_1}$ and $\...
0
votes
0answers
9 views

Separating the compact part of a Lie group

I am interested in knowing which class of Lie groups $G$ can be decomposed as a semi-direct product $$ G \sim N \rtimes H$$ where $N$ is a Lie group whose exponential is injective, and $H$ is a ...
1
vote
0answers
30 views

If $u$ is a generalized eigenvector, then $e^{tA}u = e^{t\lambda}e^{t(A-\lambda I})u$

I have come across this statement. If $\mathbf{u}$ is a generalized eigenvector, so that $(\mathbf{A}-\lambda \mathbf{I})^m \mathbf{u} = 0,$ then $$e^{t\mathbf{A}}\mathbf{u} = e^{t\lambda} e^{t(\...
0
votes
0answers
36 views

Is there a way to compute the exponential of a PDP-1 matrix?

I am computing the exponential of a matrix via Taylor expansion to prove the end-result with induction. For a matrix $A=PDP^{-1}$ where $D$ is the diagonalized matrix, is there any kind of formula ...
4
votes
4answers
155 views

Closed form expression for matrix exponential derivative with respect to scalars

I'd like to evaluate generic expressions of the following form: $$\frac{d}{da}\exp\left[aX + bY\right]$$ where $a,b$ are scalars and $X,Y$ are arbitrary complex matrices. Replacing the exponential ...
1
vote
1answer
41 views

$\int_0^t e^{sA}\cos(\omega s)ds$ with $A$ matrix

Let $A$ be a singular square matrix and $\omega,t\in\mathbb{R}^{*+}$. How to compute the following integral? $$I = \int_0^t e^{sA}\cos(\omega s)\,\mathrm{d}s$$ Since I am looking for a numerical ...
2
votes
2answers
76 views

Matrix exponential is differentiable at $0 \in \mathbb{R}^{n,n}$

Given$$\exp : \mathbb{R}^{n,n} \mapsto \mathbb{R}^{n,n} \qquad A \mapsto \sum_{k=0}^{\infty} \frac{A^k}{k!}$$ where $ \mathbb{R}^{n,n}$ is equipped with Operator Norm. I am trying to show that $\exp$ ...
0
votes
0answers
32 views

Derivative the exponential map $e^A$

Can the derivative the map $$ f(A) = e^A, A \in \mathbb{R}^{n \times n} $$ be defined as $$ \lim_{h\to 0} \frac{e^{A+h\Delta} -e^{A}}{h} =\lim_{h\to 0} e^{A} \frac{e^{h\Delta} - I}{h} $$ where $\...
2
votes
2answers
54 views

Inverse of identity minus matrix exponential

I am trying to analytically find the inverse of a matrix given by: \begin{align} W = \left( I - \alpha e^A \right)^{-1}, \end{align} where $I$ is the identity matrix of appropriate size, $e^A$ ...
0
votes
1answer
85 views

Exponential Matrix 2 - complex eigenvalues Euler

Given a matrix $A=\begin{pmatrix} \sigma & \omega \\ -\omega & \sigma \end{pmatrix}$ with two complex eigenvalues $\sigma\pm i\omega$, using the Euler formula $e^{i \omega t}=\cos(\omega t)+i \...
1
vote
1answer
54 views

The exponential of a skew-symmetric matrix in any dimension.

The Rodrigues rotation formula gives the exponential of a skew-symmetric matrix in three dimensions, and the exponential of a skew-symmetric matrix in two dimensions is given by Euler's formula. Is ...
3
votes
2answers
57 views

Power Series of an Operator

I'm working through some functional analysis problems and am having trouble with the following. Let $f(z)=\sum_{n=0}^\infty a_n z^n$ be a power series with radius of convergence $R>0$. Let $A\in\...
6
votes
2answers
464 views

If two rotation matrices commute, do their infinitesimal generators commute too?

Suppose that $e^A$ and $e^B$ are two rotations in $\mathrm{SO}(n)$. If $e^{A}e^{B} = e^{B}e^{A}$, can we conclude that $e^{A+B}=e^Ae^B$? More importantly, can we say that $AB=BA$? I'm particularly ...
1
vote
1answer
51 views

Matrix Exponential Jordan Form Linear System

Given the Linear System $\dot{x}(t)=A x(t)$ with $x_0=(x_{01},x_{02})$ as initial state and $A=\begin{pmatrix} 0 & 1 \\ -k/M & -h/M \end{pmatrix}$, when $h^2=4Mk$ the matrix A has a single ...
5
votes
1answer
125 views

If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ then $\alpha(t), \alpha'(t)$ commute?

Let $\alpha(t)$ be a smooth path of real $n \times n$ matrices. (Formally $\alpha:(-\epsilon,\epsilon) \to M_n(\mathbb{R})$). If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ for every $t$ or $(e^{\...
1
vote
0answers
20 views

Stability under conjugation of a subvector space of Lie algebra of an affine linear algebraic group (in positive characteristic)

Let $G$ a matrix Lie group and $\mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $\mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true ...
0
votes
2answers
93 views

When $e^A = e^B$ for matrices $A,B$?

Let $A$ and $B$ be $n\times n$ complex matrices such that $e^A = e^B$. I would like to know relations between $A$ and $B$. When $A,B\in\mathbb{C}$, we have a simple relation $e^A = e^B \...
1
vote
0answers
50 views

Putzer's Algorithm for matrixexponentiation

I've been looking at several different references for Putzer's Algorithm. In most cases they write the characteristic polynomial as $p_A(\lambda) = (-1)^n(\lambda-\lambda_1)(\lambda-\lambda_2)\ldots(\...
0
votes
1answer
89 views

Can we write any unitary matrix as the exponential of a skew-symmetric complex matrix?

Following the article https://en.wikipedia.org/wiki/Skew-symmetric_matrix#Infinitesimal_rotations which states that a skew symmetric (or anti-symmetric) real matrix can be written as the exponential ...
1
vote
1answer
66 views

Is there any complex number $a$ solving $\exp (X)=1+aX$ for given square matrix $X$?

I am doing exercises in Rossmann's book on Lie groups. Exercise 1.2.12 goes like this: $X \in M_n (\mathbb{C})$, $L$ is a subspace of $M_n (\mathbb{C})$, s.t. $[X, Y]\in L$ for $Y\in L$. Prove that $\...
7
votes
1answer
294 views

On the commutativity of matrices and their exponentials

It is fairly easy to see that if $A$ and $B $, real square matrices, commute, then $A $ and $e^B $ commute. In fact, $$Ae^B = \sum_{n=0}^\infty A\frac{B^n}{n!} = \sum_{n=0}^\infty \frac{B^n}{n!}A = e^...
1
vote
1answer
39 views

Matrix exponentials, computing the product of $\exp(-iBt)$ and $\exp(-iB^{-1}t)$.

Suppose I have some matrix exponential $U(t)=\exp(-iAt)$ where $t$ is some real valued number, $A$ is a hermitian matrix (so $U(t)$ is unitary) where $A=B+B^{-1}$ and $B$ is unitary. Because $B$ and $...
1
vote
1answer
28 views

matrix exponential - Is such behavior expected?

I was experimenting on $e^M$ and found this: When $ M = \left[ \begin{matrix} 0 & x \\ y & 0 \end{matrix} \right] $ $e^M = \left[ \begin{matrix} cosh(\sqrt{xy}) & \sqrt{x\over{y}}sinh(\...
6
votes
1answer
222 views

3x3 integer matrix

As far as I know, a real matrix $M$ has a real square root if $M$ is positive semidefinite, that is, if all eigenvalues are nonnegative. In fact, its square root is unique. I have read some research ...