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 R^{n} 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 λ_{1} , ..., λ_{m}, then there exist nxn symmetric idempotent matrices P_{1}, ..., P_{m} such that
The finitedimensional real spectral theorem is somewhat similar to the singularvalue decomposition theorem, which applies to all real matrices and uses two unitary matrices to accomplish a diagonalization.
The finitedimensional 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[?].
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