Questions tagged [matrix-exponential]
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"The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function."
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Is there a closed-form expression for the exponential map for SO(n), just like how Rodrigues' rotation formula is for SO(3)?
Rodrigues' rotation formula is great since it gives us a faster way to compute the exp() and log() operators for SO(3) compared to the Taylor series formulation. I was wondering if there was a ... lie-groups rotations matrix-exponential- 2,254
$e^XY$ and $Ye^X$
Let $X,Y$ be two matrices, and we define $$ e^X:=\sum_{k=0}^{\infty}\frac{1}{k!}X^k $$ In a problem about Lie algebras, I need to show if $[X,Y]=\alpha Y,\alpha\neq 2\pi ik$, then $$ e^XY=\frac{\alpha}... lie-algebras matrix-exponential- 655
Determine the image of the unit circle $S^1$ by the action of the matrix $e^A$.
We have: $$e^{ \begin{pmatrix} -5 & 9\\ -4 & 7 \end{pmatrix} }$$ I need to determine the image of the unit circle $S^1$ by the action of the matrix $e^A$. I think that I know how to calculate $... linear-algebra matrices matrix-exponential- 127
Exponential of a "simple" matrix?
I have a problem finding a simple form for $\exp(M)$ (or $\exp(tM)$), where $$M = \begin{pmatrix} 1 & a & a^2 & \dots & a^{n-1} \\ 0 & \ddots & \ddots & ... matrix-exponential- 4,654
Approximating an exponent of non-commutative matrices as a product of exponents
The book Quantum Computation and Quantum Information chapter 4.7.1 presents the following equation. \begin{equation} e^{i(A+B)\Delta t} = e^{i A \Delta t}e^{i B \Delta t} + O(\Delta t^2) \end{equation}... linear-algebra noncommutative-algebra matrix-exponential quantum-computation quantum-information- 45
Question about matrix exponentials
Below is a question and its corresponding solution involving exponentiating a matrix, which I can't understand. A boost can be written in the form $${x^\mu}^\prime=\Lambda_\nu^{{\mu}^\prime}x^\nu$$ ... matrices proof-explanation power-series hyperbolic-functions matrix-exponential- 43
Why does the matrix exponential $e^A$ always exist?
Why does $e^A$ always exist for any given $n \times n$ matrix $A$? I can't find anything discussing this question, which is quite suprising, since it is such a general question. matrices exponential-function matrix-calculus matrix-exponential- 31
Higher order derivative of exponential map
The derivative of the exponential map is given by (wiki): $$ \frac{d}{dt} e^{X(t)} = e^{X(t)} \frac{1 - e^{-ad_{X(t)}}}{ad_{X(t)}} \frac{d}{dt}X(t) $$ Is there a reasonable formula for higher order ... lie-algebras matrix-exponential- 4,488
Lie bracket for $GL_n\mathbb{R}$ from the composition of two flows of left invariant fields
I'd like to understand the following passage from Arvanitoyeorgos' "An introduction to Lie groups and the geometry of homogeneous spaces", where the author explains why for any $A,B \in M_n\... differential-geometry lie-groups lie-algebras matrix-exponential- 389
Does $e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n$ hold for matrices?
Let $X$ be a $d \times d$ real matrix, $d>1$. Is it true that $$ e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n\,\,\,? $$ Edit: It seems that this question is a duplicate. To make it ... multivariable-calculus exponential-function matrix-calculus matrix-exponential- 23.4k
Matrix exponential of infinite antisymmetric matrix with entries only next to its diagonal
What is the exponential $\exp (t A)$ of the operator $A$ whose components are given by $A_{nm} = \delta_{nm-1} \sqrt{n+1} - \delta_{nm+1}\sqrt{n}$ where the $n,m \in \mathbb{N}_0$. If we just consider ... matrices functional-analysis matrix-exponential hermite-polynomials- 133
Converting recursive equation into matrices by using matrix exponentiation
This is an example of converting fibonacci function into matrices called matrix exponentiation method. Fibonacci sequence defines $$ f(1)=1 $$ $$ f(2)=1 $$ $$ f(x) = f(x-1) + f(x-2) $$ This recursive ... linear-algebra matrices markov-chains recursive-algorithms matrix-exponential- 11
Proof the limit of matrices means about $e^H$ (where matrix $H$ is self-adjoint)
This problem is from Chap.6 of Introduction to Matrix Analysis and Applications of Petz. Prove for self-adjoint matrices $H$, $K$ that $$ \lim _{r \rightarrow 0} \left(e^{r H} \#_{\alpha}e^{r K}\... matrices matrix-calculus matrix-exponential matrix-analysis- 1
Why is the domain of the exponential function the Lie algebra and not the Lie group?
The exponential function as I know it is defined as: $$\exp:\mathfrak{g}\to G$$ and it gives each element $X$ the value of $\exp_X(1)$ where $\exp_X$ is the unique $\mathbb{R}\to G$ homomorphism that ... differential-geometry lie-groups lie-algebras matrix-exponential- 77
Why matrix exponential in two different methods not matching?
Consider the following matrix: $$A=\left[\begin{array}{ll} 1 & 1 \\ 4 & 1 \end{array}\right]$$ We need to find $e^{At}$. Method $1.$ Th eigen values of $A$ are $3,-1$. I have Diagonalized the ... matrices eigenvalues-eigenvectors matrix-calculus matrix-exponential- 9,673
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