**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 |

2 Finite dimensional case 3 See also |

## Functional Analysis

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

*j*and

*k*are distinct, and such that

*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`is the dimension of the eigenspace belonging to λ.

_{j}
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*^{*}Σ*U*

*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.