4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.
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1 4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme) Note t Cos θ Cos θ nd t t I Thus is n orthogonl mtrix. Now A t A + + θ + θ + ( ) ( ) ( ) ( ) + θ + + θ + + θ + θ + Thus if we choose θ such tht, ( + ) θ + ( )... (I) We get the entries in (,) position nd (,) position of A s zero. 5
2 (I) gives + θ + ( θ ) θ θ tn θ sgn ( ) ( ), sy..... (II) where sgn( )..... (III)..... (IV) sec θ + tn θ + from (II) + θ + θ + + 5
3 (V) + nd θ θ (VI) + (V) nd (VI) give θ, nd if we choose with these vlues of, θ, then t A A hs (,) nd (,) entries s zero. We now generlize this ide. Let A ( ij ) be n nxn rel symmetric mtrix. Let q < p < n. (Insted of (,) position bove choose (q, p) position) Consider, sgn( ) (A) 53
4 (B) (C) + θ (D) + q p O θ O θ O q p then A t A hs the entries in (q, p) position nd (p, q) position s zero. In fct A differs from A only in q th row, p th row nd q th column nd p th column nd it cn be shown tht these new entries re qi pi qi + qi pi + pi i q, p (q th row p th row)..(e) 54
5 iq ip iq + iq ip + ip i q, p (q th column p th column)..(f) θ + + θ pq θ. + θ... (G) Now the Jcobi itertion is s follows. Let A ( ij ) be nxn rel symmetric. Find q < p n such tht the off digonl entries in A. is lrgest mong the bsolute vlues of ll For this q, p find s bove. Let A t A. A cn be obtined s follows: Except the p th nd the qth rows nd the p th nd q th columns other rows nd columns of A re the sme s the corresponding rows nd columns of A, p th row, q th column, p th column which re obtined from (E), (F), (G). Now A hs in (q, p), (p, q) position. Replce A by A nd repet the process. The process converges to digonl mtrix the digonl entries of which give the eigenvlues of A. Exmple: A Entry with lrgest modulus is t (, 4) position. 55
6 q, p 4. ( ). sgn( 44 ). 4 sgn ( )( )( 4 ) ; + θ (.8944 )
7 A T A will hve 4 4. Other entries tht re different from tht of A re,, 3 ; 4, 4, 43, 44 ; (of course by symmetric corresponding reflected entries lso chnge). We hve, + 4 θ θ θ θ A Now we repet the process with this mtrix. The lrgest bsolute vlue is t (, ) position. q, p
8 gp ( ) ( 3.35)( ) sgn ; + θ The entries tht chnge re θ + θ + + θ θ.7484 nd the new mtrix is 58
9 Now we repet with q 3, p 4 nd so on. And t the th step we get the digonl mtrix giving eigenvlues of A s ,.7986, -5.64, Note: At ech stge when we choose (q, p) position nd ly the bove trnsformtion to get new mtrix A then sum of squres of off digonl entries of A will be less thn tht of A by. 59
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