First-Order Geometrical Response Equations for State ... - Jack Simons

FIRST-ORDER GEOMETRlCAL RESPONSE EQUATIONS 373formation, Eq. (49) bas to be modified by introducing extra "dummy" equations,such that the dimension of the A22block is extended to fiN x fiN and the rowdimensions of A21,vaex, and B2 ale extended to ON. In particular, by definingthe following elements:-21:::21 = ARA,nkARA,nk - { O:::22 = Ak~SBARA,SB {- l)RSl>AB(J)RZB :::2 B2RA== { ORAand the matrices:for A within the external states,for A within the internal states,for A and B both within the external states,(50a)otherwise (z is an arbitrary but nonzero constant),(50b)for A within the external states,for A within the internal states,(50c)~1l ~21+ -.A== A21 A22(A31O~~1+),A33(51a);== ~: ( vam ), "-(51b)B ~ (~),(51c)the extended set of tesponse equations can be written asAV=B. (52)The solution vector Varising from this new matrix equation can be shown tocontain the originalresponsesof Eq. (49)as wen as additionalelementsthatidentically vanish:ua = ua,v~ex= {~~::: . .vam = vam.for A within the external states,for A within the internal states,(53a)(53b)(53c)Equation(52) caDbe subjectedto a unitary transformation that eliminates theneed to know the CIvectors of aDYbut the internal states. To do sa, we first defiDeamatrix with elemeDtsURl,SA == l>RSc1. (54)