Functional Calculus
The eigendecomposition allows for much easier computation of power series of matrices. If f(x) is given by
then we know that
Because Λ is a diagonal matrix, functions of Λ are very easy to calculate:
The off-diagonal elements of f(Λ) are zero; that is, f(Λ) is also a diagonal matrix. Therefore, calculating f(A) reduces to just calculating the function on each of the eigenvalues .
A similar technique works more generally with the holomorphic functional calculus, using
from above. Once again, we find that
Read more about this topic: Eigendecomposition Of A Matrix
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