The unit process in the iterative matrix diagonalization procedure known as t me
ID: 3184084 • Letter: T
Question
The unit process in the iterative matrix diagonalization procedure known as t method is a unitary transformation that operates on rows/columns i and symmetric matrix A to make aij = aji = 0. If this transformation (from basis functions Pi and Yi to ; and ) ) is written -4.9 2aj a jj-ai (a) Show that aijis transformed to zero iftan 2- (b) Show that aan remains unchanged if neither nor v is i or j, (e Find dy and a, and show that the trace of A is not changed by thetransformation. (d) Find a and ani (where is neither i nor jand show that the sum ofthe squares of the off-diagonal elements of A is reduced by the amount 2aiExplanation / Answer
Solutions to problem 6.4.9 14/11/2017
Jacobi transformation and its properties
To make understanding easy, let me consider the case n=2 for (a) and (c)
The Jacobi transformation J is defined on a matrix A = [a ij ] as
A‘ = B = J T A J , where J is defined as rotation matrix and J = .
Substituting for J T A J , doing matrix multi plication and simplifying , we get
b 11 = a 11 c 2 – 2 c s a 12 + a 22 s 2; b22 = a11 s 2 + 2 a12 c s + a 22 c 2
b 12 = a 11 cs + a12 ( c 2– s 2) - a22 cs ; b 21 = a 11 cs + a12 ( c 2– s 2) - a22 c s
where c = cos and s = sin .
Choose b 12 = b 21 = 0
This condition gives a 11 cs + a12 ( c 2– s 2) - a22 cs =0.
ie. (a 11 – a22 ) cos sin + a12 cos 2 =0
ie. (1/2) (a 11 – a22 ) 2cos sin + a12 cos 2 =0
ie. a12 cos 2 = (1/2) (-a 11 + a22) sin
On simplification, this gives sin / cos = 2 a12 / (-a 11 + a22 )
ie. tan 2 a12 / (a 22 - a11 ), which proves (a)
The general case can be derived similarly with more computational effort.
(c)Trace A = sum of diagonal elements = (a 11 + a22 )
Trace B = b 11 + b 22 = a 11 c 2 – 2 c s a 12 + a 22 s 2 + a11 s 2 + 2 a12 c s + a 22 c 2
= a11 (c 2 + s 2 ) + a 22 (c 2 + s 2 )
= a11 + a 22 since (c 2 + s 2 ) = cos 2 + sin 2 =1
= Trace A, which proves (c)
Ie. trace is not changed by the transformation defined by J.
The solutions to (b) and (d) would be worked out separately and submitted.