Spectral theoremSpectral 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 λ1 , ..., λm, then there exist nxn hermitian idempotent operators P1, ..., Pm such that
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
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.