Spectral theorem
Spectral theorem is an important decomposition theorem of normal operators in
linear algebra and
functional analysis. The stated decomposition is called the
spectral decomposition.
If M is a normal operator, with distinct eigenvalues λ
_{1} , ..., λ
_{m}, then there exist
nxn hermitian idempotent operators
P_{1}, ...,
P_{m} such that

whenever
j and
k are distinct, and such that

The operator
P_{j} is the orthogonal projection operator whose range is that eigenspace.
In the spectral decomposition of normal matrix M, the
rank of the matrix
P_{j} 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.
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.