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Chemical Engineering Hand Written_Notes_Part_45

Chemical Engineering Hand Written_Notes_Part_45 - 92 3...

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92 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Now, a positive de fi nite symmetric matrix can be diagonalized as (5.21) B = ΨΛΨ T Where Ψ is matrix with eigen vectors as columns and Λ is the diagonal matrix with eigenvalues of B (= A T A ) on the diagonal. Note that in this case Ψ is unitary matrix ,i.e., (5.22) ΨΨ T = I or Ψ T = Ψ 1 and eigenvectors are orthogonal. Using the fact that Ψ is unitary, we can write (5.23) x T x = x T ΨΨ T x = y T y (5.24) or x T B x ( x T x ) = y T Λ y ( y T y ) where y = Ψ T x Suppose eigenvalues λ i of A T A are numbered such that (5.25) 0 λ 1 λ 2 .................. λ n Then (5.26) y T Λ y ( y T y ) = ( λ 1 y 2 1 + ................ + λ n y 2 n ) ( y 2 1 + ................. + y 2 n ) λ n This implies (5.27) y T Λ y ( y T y ) = x T B x ( x T x ) = x T ( A T A ) x ( x T x ) λ n The equality holds only at the corresponding eigenvector of A T A , i.e., (5.28) £ v ( n ) ¤ T ( A T A ) v ( n ) [ v ( n ) ] T v ( n ) = £ v ( n ) ¤ T λ n v ( n ) [ v ( n ) ] T v ( n ) = λ n Thus, (5.29) || A || 2 2 = max x 6 = 0 || Ax || 2 / || x || 2 = λ max ( A T A ) or (5.30) || A || 2 = [ λ max ( A T A )] 1 / 2 where λ max ( A T A )
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