Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESthat is, S ij =(x j ) i . Therefore A ′ is given by(S −1 AS) ij = ∑ ∑(S −1 ) ik A kl S ljk l= ∑ ∑(S −1 ) ik A kl (x j ) lk l= ∑ k= ∑ k(S −1 ) ik λ j (x j ) kλ j (S −1 ) ik S kj = λ j δ ij .So the matrix A ′ is diagonal with the eigenvalues of A as the diagonal elements,i.e.⎛λ 1 0 ··· 0⎞0 ··· 0 λ N A ′ =0 λ 2 .⎜⎝. ⎟. .. 0 ⎠ .Therefore, given a matrix A, if we construct the matrix S that has the eigenvectorsof A as its columns then the matrix A ′ = S −1 AS is diagonal and has theeigenvalues of A as its diagonal elements. Since we require S to be non-singular(|S| ≠ 0), the N eigenvectors of A must be linearly independent and form a basisfor the N-dimensional vector space. It may be shown that any matrix with distincteigenvalues can be diagonalised by this procedure. If, however, a general squarematrix has degenerate eigenvalues then it may, or may not, have N linearlyindependent eigenvectors. If it does not then it cannot be diagonalised.For normal matrices (which include Hermitian, anti-Hermitian and unitarymatrices) the N eigenvectors are indeed linearly independent. Moreover, whennormalised, these eigenvectors form an orthonormal set (or can be made to doso). Therefore the matrix S with these normalised eigenvectors as columns, i.e.whose elements are S ij =(x j ) i ,hastheproperty(S † S) ij = ∑ k(S † ) ik (S) kj = ∑ kS ∗ kiS kj = ∑ k(x i ) ∗ k(x j ) k = (x i ) † x j = δ ij .Hence S is unitary (S −1 = S † ) and the original matrix A can be diagonalised byA ′ = S −1 AS = S † AS.Therefore, any normal matrix A can be diagonalised by a similarity transformationusing a unitary transformation matrix S.286