Chemical Engineering Hand Written_Notes_Part_45

Chemical Engineering Hand Written_Notes_Part_45 - 92 3....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
92 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Now, a positive de f 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 = Ior 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_45 - 92 3....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online