Spectral theorem

The finite-dimensional real version of the spectral theorem is a theorem of linear algebra that says that any real symmetric matrix can be diagonalized by an orthogonal matrix. In other words, if M is a matrix whose entries are real numbers, and M^T=M (i.e., M is symmetric) then there is some matrix G with real entries such that G^TG=GG^T=I (i.e., G is an orthogonal matrix) and there is some real diagonal matrix D, such that

<math>G^\top MG=D</math>

or equivalently

<math>M=GDG^\top.</math>

An equivalent way of stating the theorem is in terms of eigenvalues and eigenvectors: all the eigenvalues of M are real and it is possible to find an orthonormal basis of Rⁿ consisting solely of eigenvectors of M. This orthonormal basis is given by the columns of the matrix G from above, and the diagonal matrix D contains the eigenvalues of M as entries.

Yet another way of saying the same thing is this: if M is an nxn real symmetric matrix, with distinct eigenvalues λ₁ , ..., λ_m, then there exist nxn symmetric idempotent matrices P₁, ..., P_m such that

<math>P_jP_k=0</math>

whenever j and k are distinct, and such that

<math>M=\lambda_1P_1+\cdots+\lambda_mP_m.</math>

The rank of the matrix P_j is the dimension of the eigenspace belonging to λ_j; the matrix P_j is that of the orthogonal projection operator whose range is that eigenspace. This form is the "spectral decomposition" of M, and it is generalized to an infinite-dimensional version in functional analysis.

The finite-dimensional real spectral theorem is somewhat similar to the singular-value decomposition theorem, which applies to all real matrices and uses two unitary matrices to accomplish a diagonalization.

The finite-dimensional complex version of the spectral theorem says that any normal matrix can be diagonalized by a unitary matrix, again implying that the eigenvectors are orthogonal. In that case, the matrices P_k of the previous paragraph will be Hermitian idempotent matrices rather than symmetric idempotent matrices. Furthermore, any matrix which diagonalizes in this way must be normal, hence the term normal fully characterizes matrices that are diagonalizable by unitary matrices.

Every Hermitian matrix is normal and can hence be diagonalized as above; in addition, the eigenvalues will be real in this case.

An algebraic version of the spectral theorem exists as well: a matrix M with entries in an arbitrary field can be diagonalized by some invertible matrix if and only if the minimal polynomial of M factors completely into linear terms, and the power of each linear term is 1. If the power of the linear terms is greater than one, one obtains Jordan blocks[?].