Spectral theorem

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In mathematics, the spectral theorem is an important decomposition theorem applying to normal operators in linear algebra and functional analysis. The stated decomposition is called the spectral decomposition. Since the class of normal operators includes a number of special kinds of operator (for example, a symmetric matrix is normal), the spectral theorem can be applied in a wide range of situations. In broad terms, what the spectral theorem does is to identify cases where linear operators can be modelled by multiplication operators; which are as simple as one can hope to find. See also spectral theory for historical perspective.

Table of contents

1 Finite-dimensional case

1.1 Real matrices

2 Functional analysis
3 See also

Finite-dimensional case

If M is a normal operator, with distinct eigenvalues λ₁ , ..., λ_m, then there exist 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.

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 a unitary matrix U such that

A=UΣU^*

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.

Functional analysis

A normal operator on a Hilbert space may have no eigenvalues; for example, the bilateral shift on the Hilbert space has no eigenvalues. There is a good spectral theorem for normal operators on Hilbert spaces, though, in which the sum in the finite-dimensional spectral theorem is replaced by an integral of the coordinate function over the spectrum against a projection-valued measure.

When the normal operator in question is compact, this spectral theorem reduces to the finite-dimensional spectral theorem above, except that the operator is expressed as a linear combination of possibly infinitely many projections.