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Special cases in which only complex or only real matrices are meant will be given explicit mention. 21). 3) J = diag(±1). 4) and K. 1007/978-3-642-21335-9 5, © Springer-Verlag Berlin Heidelberg 2011 39 40 5 ‘Indefinite Metric’ More generally we will allow J to be any matrix J with the property J = J −1 = J ∗ . 5) Such matrices are usually called symmetries and we will keep that terminology in our text hoping to cause not too much confusion with the term symmetry as a matrix property. In fact, without essentially altering the theory we could allow J to be just Hermitian and non-singular.

N − 1, where A(k−1) is of order k − 1 and is void for k = 1. We further partition A(n−k+1) = E C∗ , CB where E is square of order s ∈ {1, 2} and is supposed to be non-singular. For s = 1 the step is single and for s = 2 double. Set ⎡ ⎤ Ik−1 0 ⎦. X = ⎣0 0 Is −1 0 CE In−k+1−s Then ⎡ ⎤ A(k−1) 0 0 ⎦. XAX ∗ = ⎣ 0 E0 (n−k+1−s) 0 0 A In order to avoid clumsy indices we will describe the construction by the following algorithm (the symbol := denotes the common assigning operation). 5 Ψ := In ; D0 := In ; k := 1; while k ≤ n − 1 Find j such that |akj | = maxi≥k |aki |; If akj = 0 k := k + 1; End if If |akk | ≥ |akj |/2 > 0 Perform the single elimination step; A := XAX ∗ ; Ψ := X ∗ Ψ ; k := k + 1; Else If |ajj | ≥ |akj |/2 > 0 Swap the kth and the jth columns and rows in A; Swap the kth and the jth columns in Ψ ; Perform the single elimination step; A := XAX ∗ ; Ψ := X ∗ Ψ ; k := k + 1; Else Swap the k + 1th and the jth columns and rows in A; Swap the k + 1th and the jth columns in Ψ ; Perform the double elimination step; A := XAX ∗ ; Ψ := X ∗ Ψ ; k := k + 2; End if End if End while The choices of steps and swappings in this algorithm secure that the necessary inversions are always possible.

X(t) = Φ ⎣ ⎦, . 7) which is readily verified. 1). 7) is oft described by the phrase ‘any oscillation is a superposition of harmonic oscillations or eigenmodes’ which are φk (ak cos ωk t + bk sin ωk t), k = 1, . . , n. 6) in which ‘all particles oscillate in the same phase’ that is, x(t) = x0 T (t), where x0 is a fixed non-zero vector and T (t) is a scalar-valued function of t (the above formula is also well known under the name ‘Fourier ansatz’). 8) where Sj denotes any subspace of dimension j. We will here skip proving these – fairly known – formulae, valid for any pair K, M of symmetric matrices with M positive definite.