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# 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.

## 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.