What is the spectral theorem about?

The spectral theorem classifies all diagonalizable matrices in the finite-dimensional case: it shows that a matrix is diagonalizable if and only if it is a regular matrix. Note that this includes the case of self-*** yokes (ergodic). This is useful because the notion of a function f(T) (e.g., the Borel function f) of a diagonalized matrix T is clear. The usefulness of the Spectral Theorem becomes even more obvious when functions of more general matrices are used. For example, if f is analytic, its formal power series, if x is replaced by T, can be viewed as absolutely convergent in the Banach space of matrices. The spectral theorem also permits the convenient definition of the unique square root of a positive operator.

The spectral theorem can be generalized to the case of bounded regular operators on Hilbert spaces, or unbounded self-***ing conjugate operators.