Overview
Given a symmetric {% n\times n %} matrix {% A %}, the matrix exponential is defined to be
{% \displaystyle e^A = I_n + \sum_{p=1}^{\infty} \frac{A^p}{p!} %}
This is in line with the
Taylor Series
expasion of the exponential function. It is known to be absolutely convergent.
(see Gallier)
Theorems
- Given a matrix {% A %} and invertible matrix {% U %}
{% e^{UAU^{-1}} = U e^{A} U^{-1} %}
- Given any complex {% n \times n %} matrix {% A %} with eigenvalues {% \lambda_1 %}, {% \lambda_2 %},...{% \lambda_n %}, the matrix {% e^A %} has eigenvalues {% e^{\lambda_1} %},{% e^{\lambda_2} %},...{% e^{\lambda_n} %}