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.

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 , ..., λ_m, then there exist nxn hermitian idempotent operators P₁, ..., 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.

Finite dimensional case

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.

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.