The spectral theorem classifies all diagonalizable matrices in the case of finite dimensions: it shows that a matrix is diagonalizable if and only if it is normal. Please note that this includes the case of self-yoke (Hermite). This is very useful because the concept of function f(T) of diagonalized matrix T (such as Borel function f) is clear. When using more general matrix functions, the function of spectral theorem is more obvious. For example, if F is analytic, then its formal power series, if T is used instead of X, can be regarded as absolute convergence in Banach space of matrix. The spectral theorem also allows the unique square root of a positive operator to be easily defined.

The spectral theorem can be extended to the case of bounded normal operators or unbounded self-yoke operators on Hilbert space.