ChapterZero

Two ways of doing cool stuff with matrices

Filed under: Mathematics — Alex @ 5:30 pm 2/27/2007

Let be an analytic function in some domain containing the origin, with power series $f(x) = \sum_{k=0}^\infty \alpha_k x^k$ , then recall that the quantity $f(A) = \sum_{k=0}^\infty \alpha_k A^k$ is well-defined (the series converges) if you can find some matrix norm in which $f(\|A\|) = \sum_{k=0}^\infty \alpha_k \|A\|^k$ 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 $\lambda_1, \ldots, \lambda_k$ then there is a partition of unity $I = \sum_{i=1}^k A_{i}$ where the A_i is the orthogonal projection unto the -th eigenspace of ; this partition satisfies $A = \sum_{i=1}^k \lambda_i A_i$ . Then we define the application of an analytic function to as $f(A) = \sum_{i=1}^k f(\lambda_i) A_i$ .

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) $A_i A_j = A_i \delta_{ij}$ and A_i^n = A_i strongly suggest that if you put the spectral decomposition into the standard definition and expanded $f(\sum_{i=1}^k \lambda_i A_i)$ 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 A_i is much more easily systematized and quicker than the calculation of the Jordan blocks, I think this is a pretty useful fact.

Two ways of doing cool stuff with matrices

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