ChapterZero

Let be an analytic function in some domain containing the origin, with power series , then recall that the quantity is well-defined (the series converges) if you can find some matrix norm in which converges.

That’s the ’standard’ definition of a matrix function, as I’ve known it. But we learned a pretty neat definition in our linear algebra class recently, which has more of the feel of a definition than a generalization. When both definitions apply to a given matrix, the result *seems* to be the same. The algebraic definition depends on the spectral decomposition theorem– a pretty cool result in itself–, which says that if is a Hermitian matrix with unique eigenvalues then there is a partition of unity where the is the orthogonal projection unto the -th eigenspace of ; this partition satisfies . Then we define the application of an analytic function to as .

I have to check that the results of the two definitions are the same, but I suspect (strongly enough not to bother checking ) that they are, because of the mutual requirement for analyticity (why else would an analytic constraint be given in an algebraic definition?); beyond that, the fact(s) and strongly suggest that if you put the spectral decomposition into the standard definition and expanded you’d get the result of the linear algebraic definition.

Practically, this means if you’re computing matrix exponentials say, and you have the good fortune of dealing with a normal matrix, you can avoid getting its Jordan form by using the linear algebraic definition. Considering that the calculation of the is much more easily systematized and quicker than the calculation of the Jordan blocks, I think this is a pretty useful fact.