Ivancevic_Applied-Diff-Geom

1032 Applied Differential Geometry: A Modern Introductionextract from the argument of the exponential function a quadratic formn+1∑[y k − y k−1 ] 2 = y0 2 − 2y 1 y 0 + y1 2 + y1 2 − 2y 1 y 2 + . . . + yn+12k=1= y t My + [y 2 0 − 2y 1 y 0 + y 2 n+1 − 2y n y n+1 ], (6.49)by introducing the nD array y and the nxn matrix M defined as [Montagnaet. al. (2002)]⎛ ⎞⎛⎞y 12 −1 0 · · · · · · 0y 2−1 2 −1 0 · · · 0y =.0 −1 2 −1 · · · 0, M =, (6.50)⎜ ⎟0 · · · −1 2 −1 0⎝ . ⎠⎜⎟⎝ 0 · · · · · · −1 2 −1 ⎠y n 0 · · · · · · · · · −1 2where M is a real, symmetric, non singular and tridiagonal matrix. Interms of the eigenvalues m i of the matrix M, the contribution in (6.49) canbe written asy t My = w t O t MOw = w t M d w =n∑m i wi 2 , (6.51)by introducing the orthogonal matrix O which diagonalizes M, with w i =O ij y j . Because of the orthogonality of O, the Jacobian∣ J = detdw i ∣∣∣∣ = det |O ki |,dy kof the transformation y k → w k equals 1, so that ∏ ni=1 dw i = ∏ ni=1 dy i.After some algebra, (6.49) can be written asn+1∑[y k − y k−1 ] 2 =k=1i=1n∑m i wi 2 + y0 2 − 2y 1 y 0 + yn+1 2 − 2y n y n+1 =i=1n∑i=1[m i w i − (y ] 20O 1i + y n+1 O ni )+ y0 2 + yn+1 2 −m in∑i=1(y 0 O 1i + y n+1 O ni ) 2m i.(6.52)