Chemical Engineering Hand Written_Notes_Part_61

Chemical Engineering Hand Written_Notes_Part_61 - 124 3...

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124 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES (24) Show that vector x y is orthogonal to vector x + y if and only if || x || = || y || . (25) For a positive de fi nite matrix A, the Cholensky decomposition is A = L D L T = RR T where R = LD 1 / 2 . Show that the condition number of R is square root of condition number of A. It follows that Gaussian elimination needs no row exchanges for a positive de fi nite matrix; the condition number does not deteriorate, since c ( A ) = c ( R T ) c ( R ) . (26) Show that for a positive de fi nite symmetric matrix, the condition num- ber can be obtained as c ( A ) = λ max ( A ) min ( A ) (27) If A is an orthonormal(unitary) matrix (i.e. A T A = I ) , show that || A || = 1 and also c ( A ) = 1 . Orthogonal matrices and their multiples ( αA )are he only perfectly conditioned matrices. (28) Show that λ max (i.e. maximum magnitude eigen value of a matrix) or even max | λ i | , is not a satisfactory norm of a matrix, by fi nding a 2x2 counter example to λ max ( A + B ) λ max ( A ) + λ max ( B ) and to λ max ( AB ) λ max ( A ) λ max ( B )
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