Lecture 13 Eigenvalue Problems

Transcription

1 Lecture 13 Eigenvalue Probles MIT J / 6.337J Introduction to Nuerical Methods Per-Olof Persson October 24,

2 The Eigenvalue Decoposition Eigenvalue proble for atrix A: Ax = λx with eigenvalues λ and eigenvectors x (nonzero) Eigenvalue decoposition of A: A = XΛX 1 or AX = XΛ with eigenvectors as coluns of X and eigenvalues on diagonal of Λ In eigenvector coordinates, A is diagonal: Ax = b (X 1 b) = Λ(X 1 x) 2

3 Multiplicity The eigenvectors corresponding to a single eigenvalue λ (plus the zero vector) for an eigenspace Diension of E λ = di(null(a λi)) = geoetric ultiplicity of λ The characteristic polynoial of A is p A (z) = det(zi A) = (z λ 1 )(z λ 2 ) (z λ ) λ is eigenvalue of A p A (λ) = 0 Since if λ is eigenvalue, λx Ax = 0. Then λi A is singular, so det(λi A) = 0 Multiplicity of a root λ to p A = algebraic ultiplicity of λ Any atrix A has eigenvalues, counted with algebraic ultiplicity 3

4 Siilarity Transforations The ap A X 1 AX is a siilarity transforation of A A and B are siilar if there is a siilarity transforation B = X 1 AX A and X 1 AX have the sae characteristic polynoials, eigenvalues, and ultiplicities: The characteristic polynoials are the sae: p X 1 AX(z) = det(zi X 1 AX) = det(x 1 (zi A)X) = det(x 1 )det(zi A)det(X) = det(zi A) = p A (z) Therefore, the algebraic ultiplicities are the sae If E λ is eigenspace for A, then X 1 E λ is eigenspace for X 1 AX, so geoetric ultiplicities are the sae 4

5 Algebraic Multiplicity Geoetric Multiplicity Let n first coluns of Vˆ be orthonoral basis of the eigenspace for λ Extend Vˆ to square unitary V, and for λi C B = V AV = 0 D Since det(zi B) = det(zi λi)det(zi D) = (z λ) n det(zi D) the algebraic ultiplicity of λ (as eigenvalue of B) is n A and B are siilar; so the sae is true for λ of A 5

6 Defective and Diagonalizable Matrices If the algebraic ultiplicity for an eigenvalue > its geoetric ultiplicity, it is a defective eigenvalue If a atrix has any defective eigenvalues, it is a defective atrix A nondefective or diagonalizable atrix has equal algebraic and geoetric ultiplicities for all eigenvalues The atrix A is nondefective A = XΛX 1 ( =) If A = XΛX 1, A is siilar to Λ and has the sae eigenvalues and ultiplicities. But Λ is diagonal and thus nondefective. (= ) Nondefective A has linearly independent eigenvectors. Take these as the coluns of X, then A = XΛX 1. 6

7 Deterinant and Trace The trace of A is tr(a) = j=1 a jj The deterinant and the trace are given by the eigenvalues: det(a) = λ j, tr(a) = λ j j=1 j=1 since det(a) = ( 1) det( A) = ( 1) p A (0) = j=1 λ j and p A (z) = det(zi A) = z j=1 p A (z) = (z λ 1 ) (z λ ) = z 1 a jj z + 1 λ j z + j=1 7

8 Unitary Diagonalization and Schur Factorization A atrix A is unitary diagonalizable if, for a unitary atrix Q, A = QΛQ A heritian atrix is unitarily diagonalizable, with real eigenvalues (because of the Schur factorization, see below) A is unitarily diagonalizable A is noral (A A = AA ) Every square atrix A has a Schur factorization A = QTQ with unitary Q and upper-triangular T Suary, Eigenvalue-Revealing Factorizations Diagonalization A = XΛX 1 (nondefective A) Unitary diagonalization A = QΛQ (noral A) Unitary triangularization (Schur factorization) A = QTQ (any A) 8

9 Eigenvalue Algoriths The ost obvious ethod is ill-conditioned: Find roots of p A (λ) Instead, copute Schur factorization A = QTQ by introducing zeros However, this can not be done in a finite nuber of steps: Any eigenvalue solver ust be iterative To see this, consider a general polynoial of degree p(z) = z + a 1 z a 1 z + a 0 There is no closed-for expression for the roots of p: (Abel, 1842) In general, the roots of polynoial equations higher than fourth degree cannot be written in ters of a finite nuber of operations 9

10 Eigenvalue Algoriths (continued) However, the roots of p are the eigenvalues of the copanion atrix 0 a0 1 0 a a 2 A = a 2 1 a 1 Therefore, in general we cannot find the eigenvalues of a atrix in a finite nuber of steps (even in exact arithetic) In practice, algoriths available converge in just a few iterations 10

11 Schur Factorization and Diagonalization Copute Schur factorization A = QTQ by transforing A with siilarity transforations which converge to a T as j Q j Q 2Q 1 A Q 1 Q 2 Q j }{{}}{{} Q Q Note: Real atrices ight need coplex Schur fors and eigenvalues (or a real Schur factorization with 2 2 blocks on diagonal) For heritian A, the sequence converges to a diagonal atrix 11

12 Two Phases of Eigenvalues Coputations General A: First to upper-hessenberg for, then to upper-triangular Phase 1 Phase 2 A = A H T Heritian A: First to tridiagonal for, then to diagonal Phase 1 Phase 2 A = A T D 12