lecture_24 (dragged) 2

lecture_24 (dragged) 2 - MA 36600 LECTURE NOTES: FRIDAY,...

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MA 36600 LECTURE NOTES: FRIDAY, MARCH 13 3 In general, the determinant of an n × n matrix is ° ° ° ° ° ° ° ° ° ° ° a 11 a 21 ··· a n 1 a 12 a 22 ··· a n 2 . . . . . . . . . . . . a 1 n a 2 n ··· a nn ° ° ° ° ° ° ° ° ° ° ° = ± σ ° ( σ ) · a σ (1)1 a σ (2)2 ··· a σ ( n ) n , where the sum is taken over all n ! permutations σ of the set { 1 , 2 ,...,n } , and ° ( σ )= ± 1 is the “sign” of the permutation. In particular, when a ij = y ( j 1) i , we have the expression W = ± σ ° ( σ ) · y (0) σ (1) y (1) σ (2) ··· y ( n 1) σ ( n ) = ± σ ° ( σ ) · n ² j =1 y ( j 1) σ ( j ) . In order to take the derivative, we note the chain rule extended for the product of n functions: f = n ² j =1 f i = ln f = n ± j =1 ln f j = f (1) f = n ± k =1 f (1) k f k = f (1) = n ± k =1 f (1) k ² j ° = k f j . Hence we Fnd that W (1) = ± σ ° ( σ ) · n ± k =1 y ( k ) σ ( k ) ² j ° = k y ( j 1) σ ( j ) = n ± k =1 ± σ ° ( σ ) · y ( k ) σ ( k ) ² j ° = k y ( j 1) σ ( j ) = n ± k =1 ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° °
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