6, Eigenvalues and Eigenvectors. = λx,

Transcription

1 6, Eigenvalues and Eigenvectors The eigenvalue/eigenvector of an N N matrix A are defined as a scalar λ and a nonzero vector v satisfying A x = λx, (A λi)v =, (v ). the matrix [A λi] should be sigular, its determinant should be zero det(a λi) =, the determinant of the matrix [A λi] is a polynomial of degree N in terms of λ. IPT-526, Spring 26 p./25

2 Eigenvalues and Eigenvectors. find the eigenvalue λ s by solving the characteristic equation, A λi = λ N + a N λ N + + a λ + a =, 2. then substitute the λ i s, one by one, into (A λi)v =, to solve it for the eigenvector v i s. IPT-526, Spring 26 p.2/25

3 Scaled power method: Largest magnitude Suppose all of the eigenvalues of an N N matrix A are distinct with the magnitudes, λ > λ 2 λ 3 λ N. For any initial vector x, such as x = [ ], consequently, any initial vector x can be expressed as a linear combination of the eigenvecotrs, For Av n = λ n v n,we have x = α v + α 2 v α N v N. Ax = α λ v + α 2 λ 2 v α N λ N v N = λ (α v + α 2 λ 2 λ v α N λ N λ v N ), IPT-526, Spring 26 p.3/25

4 Scaled power method: Largest magnitude repeat this multiplication over and over again to obtain, x k = A k x = λ k (α v + α 2 ( λ 2 λ ) k v α N ( λ N λ ) k v N ) λ k α v, we must keep scaling before multiplying at every iteration to prevent the overflow or underflow, where x = Max{ x n }. x k+ = A x k x k λ v v, IPT-526, Spring 26 p.4/25

5 Inverse power method: smallest magnitude If the matrix is nonsigular A, then A x = λx, A x = λ x, this implies that the inverse matrix A has the eigenvalues that are the reciprocals of the eigenvalues of the original matrix A, but with the same eigenvectors. we can apply the scaled power method for the eigenvalues with largest magnitude, λ N = the largest eigenvalue of A, to find the smallest (magnitude) eigenvalue λ N. IPT-526, Spring 26 p.5/25

6 Shifted inverse power method To find the eigenvalue that is not necessarily of the largest or smallest magnitude, we substract the eigenvalue equation by sv (s is a number that does not happen to equal any eigenvalue,) A x = λx, [A si]x = (λ s)x, this implies that (λ s) is the eigenvalue of [A si], we apply the inverse power method to get its smallest magnitude eigenvalue, then we add s to obtain the eigenvalue of the original matrix A which is closest to the number s, λ N = the largest eigenvalue of [A si] + s, to find the smallest (magnitude) eigenvalue λ N. IPT-526, Spring 26 p.6/25

7 Eigenpackages of Canned eigenroutines LAPACK: Linear Algebra Package EISPACK: Matrix Eigensystem Routines IMSL: Visual Numerics NAG: Numerical Algorithms Group IPT-526, Spring 26 p.7/25

8 Eigenpackages of Canned eigenroutines all eigenvalues and no eigenvectors, all eigenvalues and some corresponding eigenvectors, all eigenvalues and all corresponding eigenvectors, real, symmetric, tridiagonal, real, symmetric, banded, real, symmetric, full, real nonsymmetric, complex, Hermitian, complex, non-hermitian. IPT-526, Spring 26 p.8/25

9 Gap solitons in optical lattices -D Gross-Pitaevskii equation with periodic potentials, V (x) = V sin 2 (k x), i t Φ (t, x) = 2 2 x 2 Φ (t, x) + V (x)φ (t, x) + g D Φ (t, x) 2 Φ (t, x) which has gap soliton solutions. 2 2 (a) 8 c - (b) P 6 4 b - (c) 2 a µ X by FS + NK methods, N x = 52, no. of iterations < ; gpeol.m. R.-K. Lee, E. A. Ostrovskaya, Yu. S. Kivshar, and Y. Lai, Phys. Rev. A 72, 3367 (25). IPT-526, Spring 26 p.9/25

10 Band-Gap spectrum -D Gross-Pitaevskii equation with periodic potentials, V (x) = V sin 2 (k x), i t Φ (t, x) = 2 2 x 2 Φ (t, x) + V (x)φ (t, x) + g D Φ (t, x) 2 Φ (t, x). For the stationary solution, Φ (t, x) = φ(x)exp( iµt), one have time-independent GP equation, 2 d 2 φ dx 2 [V sin 2 (Kx)φ µφ] g D φ 2 φ =. For the linear case (non-interacting condensate), g D =, 2 d 2 φ [2q cos(2y) p]φ =, dy2 where y = Kx, q = V /(2K 2 ), and p = 2(q + µ/k 2 ), which is the Mathieu s equation. IPT-526, Spring 26 p./25

11 Mathieu equation Mathieu equation: u xx + 2q cos(2x)u = λu, λ q IPT-526, Spring 26 p./25

12 Nonlinear band structure for BEC in OL 2 d 2 φ dx 2 [V sin 2 (Kx)φ µφ] σ φ 2 φ =. M. Machholm et al., Phys. Rev. A 67, 5363 (23). IPT-526, Spring 26 p.2/25

13 Bragg reflectors A B A B A B A B Transmission δ (/cm) Nobel laureates William Henry Bragg and William Lawrence Bragg in 95, for their contribution to the X-ray diffraction analysis (Bragg diffraction). IPT-526, Spring 26 p.3/25

14 Photonic Bandgap Crystals: two(high)-dimension a =.45 2r =.5 d =.45 l =. n c = Transmittance [db] # of layers = Normalized Frequency ωa/2πc IPT-526, Spring 26 p.4/25

15 Band diagram and Density of States Frequency (ωd/2πc) M Γ X M DOS (A.U.) x 4 IPT-526, Spring 26 p.5/25

16 Photonic Bandgap Crystals:point-defect (localized field) Transmittance [db] Normalized Frequency ω a/2πc IPT-526, Spring 26 p.6/25

17 Photonic Bandgap Crystals:line-defects Courtesy: Bin-Shei Lin IPT-526, Spring 26 p.7/25

18 Maxwell equations For a linear, isotropic nondispersive material, B = µh, D = ǫe. magnetic conduction current: J m = ρ H. electric conduction current: Then the Maxwell s curl equations becom J e = σe. t H = µ E ρ µ H t E = ǫ H σ ǫ E. IPT-526, Spring 26 p.8/25

19 For a periodic structure, Periodic structrures D(r,t) = ǫ ǫ(r)e(r,t) the periodicity of ǫ(r) implies ǫ(r + a i ) = ǫ(r), (i =, 2, 3). then the Maxwell equations become, {ǫ(r)e(r,t)} =, H(r,t) =, E(r,t) = µ t H(r,t), H(r,t) = ǫ ǫ(r) t E(r,t). IPT-526, Spring 26 p.9/25

20 Wave equations and plane wave expansion then one can obtain the wave equations, ǫ(r) { E(r, t)} = 2 c 2 E(r, t), t2 { ǫ(r) H(r, t)} = 2 c 2 H(r, t), t2 use plane wave solution as basis, E(r, t) = E(r)e iωt, H(r, t) = H(r)e iωt, then the wave equations become the eigenvalue equations, L E E(r) = ǫ(r) L H H(r) = { ǫ(r) ω2 { E(r, t)} = E(r, t), c2 ω2 H(r, t)} = H(r, t), c2 IPT-526, Spring 26 p.2/25

21 Brillouin zone For a wave vector k in the first Brillouin zone and a band index n, we can express the field as, E(r) = E kn (r) = u kn (r)e ik r, where u kn (r) is a periodic function satisfying, u kn (r + a i ) = u kn (r). only consider the E-field in two-dimensional space, ǫ(r) { E(r)} = ω2 c 2 E(r), which can be reduced to L (2) E E z(r) = ǫ(r) { 2 x y 2 }E z(r) = ω2 c 2 E(r), IPT-526, Spring 26 p.2/25

22 Brillouin zone apply Bloch s theorem, E z (r) = E z,kn (r) = G E z,kn (G)exp{i(k + G) r}, wher k and G are the reciprocal lattice vector in two dimensions, G = l b + l 2 b 2, where a i b j = 2πδ ij and {l i } are arbitrary integers. the final eigenvalue equation to solve is, ǫ(g G ) k + G 2 E z,kn (G ) = ω2 kn c E z,kn(g ). 2 G IPT-526, Spring 26 p.22/25

23 Band diagram and Density of States ω2 { E(r)} = ǫ(r) c E(r), Frequency (ωd/2πc) M Γ X M DOS (A.U.) x 4 IPT-526, Spring 26 p.23/25

24 Laplacian equation in a disk Eigenmodes of Laplacian equations, [ 2 x x 2 ]u(x, y) = λf(x, y). Mode λ =. Mode 3 λ = Mode 6 λ = Mode λ = L. N. Trefethen, Spectral methods in Matlab, SIAM (2). IPT-526, Spring 26 p.24/25

25 Nonlinear eigenvalue problems For example, (Aλ 2 + Bλ + C) x =, introduce an additional unkown eigenvector y and solve the 2N 2N eigensystem, x = λ x A C A B y y This technique generalizes to higher-order polynomials in λ. Apolynomial of degree M produces a linear MN MN eigensystem. IPT-526, Spring 26 p.25/25