Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES◮Construct an orthonormal set of eigenvectors for the matrix⎛A = ⎝ 1 0 30 −2 0⎞⎠ .3 0 1We first determine the eigenvalues using |A − λI| =0:1 − λ 0 30=0 −2 − λ 0∣ 3 0 1− λ ∣ = −(1 − λ)2 (2 + λ) + 3(3)(2 + λ)=(4− λ)(λ +2) 2 .Thus λ 1 =4,λ 2 = −2 =λ 3 . The eigenvector x 1 = (x 1 x 2 x 3 ) T is found from⎛⎝ 1 0 3⎞ ⎛0 −2 0 ⎠ ⎝ x ⎞ ⎛1x 2⎠ =4⎝ x ⎞⎛1x 2⎠ ⇒ x 1 = √ 1 ⎝ 1 ⎞0 ⎠ .3 0 1 x 3 x 32 1A general column vector that is orthogonal to x 1 isx = (a b − a) T , (8.89)and it is easily shown that⎛⎞ ⎛ ⎞ ⎛ ⎞Ax = ⎝ 1 0 30 −2 0 ⎠ ⎝a b ⎠ = −2 ⎝a b ⎠ = −2x.3 0 1 −a−aThus x is a eigenvector of A with associated eigenvalue −2. It is clear, however, that thereis an infinite set of eigenvectors x all possessing the required property; the geometricalanalogue is that there are an infinite number of corresponding vectors x lying in theplane that has x 1 as its normal. We do require that the two remaining eigenvectors areorthogonal to one another, but this still leaves an infinite number of possibilities. For x 2 ,therefore, let us choose a simple form of (8.89), suitably normalised, say,x 2 = (0 1 0) T .The third eigenvector is then specified (to within an arbitrary multiplicative constant)by the requirement that it must be orthogonal to x 1 and x 2 ; thus x 3 may be found byevaluating the vector product of x 1 and x 2 and normalising the result. This givesx 3 = 1 √2(−1 0 1) T ,to complete the construction of an orthonormal set of eigenvectors. ◭8.15 Change of basis and similarity transformationsThroughout this chapter we have considered the vector x as a geometrical quantitythat is independent of any basis (or coordinate system). If we introduce a basise i , i =1, 2,...,N, into our N-dimensional vector space then we may writex = x 1 e 1 + x 2 e 2 + ···+ x N e N ,282