can always be factored into linear factors. For an arbitrarynbyncomplex ma-
trix there will bensolutions (counting repeated eigenvalues with multiplicity).
A basis will exist for which the matrix will be in upper triangular form.
The case of self-adjoint matricesLis much more constrained, since transpo-
sition relates matrix elements. One has:
Theorem 4.1(Spectral theorem for self-adjoint matrices).Given a self-adjoint
complexnbynmatrixL, there exists a unitary matrixUsuch that
ULU−^1 =DwhereDis a diagonal matrix with entriesDjj=λj,λj∈R.
GivenL, its eigenvaluesλjare the solutions to the eigenvalue equation 4.5 and
Uis determined by the eigenvectors. For distinct eigenvalues the corresponding
eigenvectors are orthogonal.
This spectral theorem here is a theorem about finite dimensional vector
spaces and matrices, but there are analogous theorems for self-adjoint operators
on infinite dimensional state spaces. Such a theorem is of crucial importance in
quantum mechanics, where forLan observable, the eigenvectors are the states
in the state space with well-defined numerical values characterizing the state,
and these numerical values are the eigenvalues. The theorem tells us that, given
an observable, we can use it to choose distinguished orthonormal bases for the
state space by picking a basis of eigenvectors, normalized to length one.
Using the bra-ket notation in this case we can label elements of such a basis
by their eigenvalues, so
|j〉=|λj〉
(theλjmay include repeated eigenvalues). A general state is written as a linear
combination of basis states
|ψ〉=∑
j|j〉〈j|ψ〉which is sometimes written as a “resolution of the identity operator”
∑
j|j〉〈j|= 1 (4.6)Turning from self-adjoint to unitary matrices, unitary matrices can also be
diagonalized by conjugation by another unitary. The diagonal entries will all be
complex numbers of unit length, so of the formeiλj,λj∈R. For the simplest
examples, consider the cases of the groupsSU(2) andU(2). Any matrix inU(2)
can be conjugated by a unitary matrix to the diagonal matrix
(
eiλ^10
0 eiλ^2
)
which is the exponential of a corresponding diagonalized skew-adjoint matrix
(
iλ 1 0
0 iλ 2