In
mathematics, particularly
linear algebra and
functional analysis, the
spectral theorem is any of a number of results about
linear operators or
matrices. In broad terms, the spectral
theorem provides conditions under which an
operator or a matrix can be
diagonalized (that is, represented as a
diagonal matrix in some basis). Intuitively, diagonal matrices are computationally quite manageable, so it is of interest to see whether an arbitrary matrix can be diagonalized. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of
linear operators that can be modeled by
multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative
C*-algebras. See also
spectral theory for a historical perspective.