Spectral Analysis of Matrices - An Introduction for Engineers

Transcription

1 Spectral Analysis of Matrices - An Introduction for Engineers Timo Weidl Department of Mathematics Royal Institute of Technology S-144 Stockholm weidl@mathkthse January 13, 26

2 Plan of the Talk 1 A practical problem: Stable rotation of a three-dimensional body 2 Eigenvalues and eigenvectors of matrices 3 The characteristic polynomial 4 Some examples and problems 5 A practical problem: Torsion oscillations of a gear shaft

3 1 A practical problem: Stable rotation of a three-dimensional body Consider a stiff body Ω with the density ρ(x) Chose the system of coordinates, which is fixed with the body at the centre of mass The description of the rotation of Ω invokes the matrix of the moments of inertia: t kl = δ kl = T = 3 δ kl Ω j=1 1 if k = l if k l t 11 t 12 t 13 t 21 t 22 t 23 t 31 t 32 t 33, x 2 j x lx k ρ(x)d 3 x,, k, l = 1, 2, 3 Let w be the angular velocity of Ω This is a vector, directed along the axes of the rotation

4 The energy E and the spin s are given by E = 1 2 < T w, w >= 1 2 < s, w > and s = T w A body can rotate freely around the axes along w, if and only if s and w are parallel: λw = s = T w Mathematical problem: Find all λ and w, such that T w = λw () 2 Eigenvalues and eigenvectors of linear operators Consider the finite-dimensional complex vector space H = C n Let 1 denote the idendity on H Consider a linear operator T on H Definition 1 Assume that for some λ C and some w C n, w, the identity T w = λw (2)

5 is satisfied Then λ is said to be an eigenvalue of T and w is the respective eigenvector Definition 2 Let λ be an eigernvalue of T The eigenspace E λ of T corresponding to λ is defined as the set of all solutions w C n of (2) Remark An eigenspace E λ of T is a linear subspace of H = C n Indeed, for v, w E λ and α, β C we have T (αv + βw) = αt v + βt w = αλv + βλw = λ(αv + βw) and hence αv + βw E λ Definition 3 The set of all eigenvalues of T is called the spectrum σ(t ) of T Consider the operator function R T (λ) = (T λ 1) 1 It is well defined, iff the equation (T λ1)w = does not have a non-trivial solution, that is iff λ σ(t )

6 Definition 4 The set ρ(t ) = C\σ(T ) is called the resolvent set, and the operator function R T (λ) is called the resolvent of T 3 The characteristic polynomial Definition 5 The function d T (λ) : = det(t λ 1) = det t 11 λ t 12 t 1n t 21 t 12 λ t 1n t n1 t n2 t nn λ is called the characteristic polynomial of T = Remark The roots of d T (λ) are the eigenvalues of T Problem: Show that d T (λ) is a polynomial of degree n! Remark The spectrum σ(t ) consists of at least one and of not more than n points The polynomial d T (λ) can be factorized as follows: d T (λ) = (λ λ 1 ) τ 1 (λ λ l ) τ l,

7 where λ l are the roots of order τ l N We have τ τ l = n, and hence l n We set d T (λ) = a n λ n + + a 1 λ + a Problem: Determine a n, a n 1 and a! Definition 6 The order τ k of the root λ k of d T (λ) is called the algebraic multiplicity of the eigenvalue λ k of T Definition 7 The number ν k = dim E λk is called the geometric multiplicity of the eigenvalue λ k of T Proposition 1 It holds (Proof follows later) ν k τ k for all k = 1,, l 4 Some examples and problems ( ) 1 2 For T = we have 2 1 ( 1 λ 2 d T (λ) = det 2 1 λ ) = (1 λ)(1 λ) + 4 = λ 2 2λ + 5 = (1 + 2i λ)(1 2i λ)

8 Hence T has the eigenvalues λ 1 = 1 2i, λ 2 = 1 + 2i with the multiplicities ν 1 = ν 2 = τ 1 = τ 2 = 1 eigenvectors w (1) corresponding to λ 1 solve ( 1 λ λ 1 We see that w (1) = β ) w(1) 1 w (1) 2 ) ( i 1 ( 2i 2 = 2 2i, β C ) w(1) 1 w (1) 2 The = Determine the eigenspace E λ2! In higher dimensions n > 4 the roots of the characteristic polynomial can usually not be found explicitly Therefore other techniques are required to calculate or estimate the spectrum Let us consider the special case of diagonal matrices Put T = t 1 t 2 t 3 t n

9 Determine d T (λ), σ(t ), ρ(t ) and R T (λ)! Let T be some linear operator on C n and let A be an invertible linear operator Put S = A 1 T A Show that σ(s) = σ(t )! 5 A practical problem: Torsion oscillations of a gear shaft Consider a gear shaft consisting of n wheels with the moments of inertia m 1,, m n > w 1 (t) Let w(t) = be the vector of the torsion angles w n (t) with respect to the normal position at the time t The torque acting on the l-th wheel is proportional to the relative torsion of the two near-by wheels m l ẅ l = (w l 1 w l )k l (w l w l+1 )k l+1, l = 1,, n, w := w 1, w n+1 := w n

10 Let K be the matrix K = and let M be the matrix Then k 2 k 2 k 2 k 2 k 3 k 3 k 3 k 3 k 4 k 4 M = m 1 m 2 m 3 m n Mẅ(t) = Kw(t) We look for an oscillating solution of the type w(t) = e iκt v, v = 1 v that is with λ = κ 2 for a solution of the v n, Mathematical problem: Find all λ and v, such that λmv = Kv λv = (M 1 K)v (1)