Main Page | See live article | Alphabetical index Spectral theorem is an important decomposition theorem of normal operators in linear algebra and functional analysis. The stated decomposition is called the spectral decomposition. Table of contents 1 Functional Analysis 2 Finite dimensional case 2.1 Real matrices 3 See also Functional Analysis If M is a normal operator, with distinct eigenvalues λ1 , ..., λm, then there exist nxn hermitian idempotent operators P1, ..., Pm such that whenever j and k are distinct, and such that The operator Pj is the orthogonal projection operator whose range is that eigenspace. Finite dimensional case In the spectral decomposition of normal matrix M, the rank of the matrix Pj is the dimension of the eigenspace belonging to λ. A more familiar form of spectral theorem is that any normal matrix can be diagonalized by a unitary matrix. That is, for any normal matrix A, there exists an unitary matrix U such that A=U*ΣU where Σ is the diagonal matrix where the entries are the eigenvalues of A. Furthermore, any matrix which diagonalizes in this way must be normal. The column vectors of U are the eigenvectors of A and they are orthogonal. It could be viewed as a special case of Schur decomposition. Real matrices If A is a real symmetric matrix, then U could be chosen to be an orthogonal matrix and all the eigenvalues of A are real. See also Matrix decomposition Jordan decomposition, an "algebraic" analogue to spectral decomposition. Singular value decomposition, a generalisation of spectral theorem to arbitrary matrices. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.