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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.

Table of contents
1 Functional Analysis
2 Finite dimensional case
3 See also

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

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.

See also