Archive for April, 2011

Matrix Exponentials

April 2, 2011

A PDF version of this post (along with the LaTeX file) is available HERE.

For any element a of any finite dimensional of any finite dimensional \mathbb C-algebra with 1, let f_a\in\mathbb C[X] be the unique polynomial of degree <\dim\mathbb C[a] satisfying f_a(a)=e^a. (The letter X is an indeterminate.)

Let m\in\mathbb C[X] be the minimal polynomial of a, let \lambda be a multiplicity \mu(\lambda) root of m, and let x(\lambda) be the image of X in \mathbb C[X]/(X-\lambda)^{\mu(\lambda)}.

f_a\equiv f_{x(\lambda)}\quad\bmod\quad(X-\lambda)^{\mu(\lambda)},

where \lambda runs over the roots of m.

Here are some more details:

\displaystyle f_{x(\lambda)}=e^\lambda\ \sum_{n<\mu(\lambda)}\ \frac{(X-\lambda)^n}{n!}\quad,

\displaystyle f_a=\sum_\lambda\ T_\lambda\left(f_{x(\lambda)}\ \frac{(X-\lambda)^{\mu(\lambda)}}{m}\right)\frac{m}{(X-\lambda)^{\mu(\lambda)}}\quad,

where T_\lambda means “degree <\mu(\lambda) Taylor polynomial at X=\lambda”.

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