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matlab讲义4.ppt

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matlab讲义4.ppt
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Week 4,DeterminantInner Products,Given A = [aij],Where Mij is the submatrix obtained by deleting row i and column j.,Why compute det(A)?,det(A) = 0  A is singular,Determinant,det(A) = ad - bc,For 3-by-3 matrix,det(A) = aei + dhc + gbf – ceg – fha – ibd,Not true for general n-by-n matrix.,,For 2-by-2 matrix,If n = 20, n! = 2  1018 operations. Assuming 109 operations/sec.  2  109 sec. = 100yr. So how det(A) is computed? Find it by Matlab!,Wn ≑ n Wn-1≑ n • (n-1) • Wn-2≑ n! W1 = n! 1 = n!,If work of computing det(An) = Wn, then,n=500; A=rand(n); % generate a n-by-n matrix tic; % set the clock to zero v=det(A); % compute the determinant toc % show the elapsed time,To determine the timing,n time ratio0.1090 ---0.7350 6.74 2000 5.8440 7.95 4000 43.5000 7.44,Conclusion: det(An) = O(n3),Let V be a vector space. A function F: V VC is an inner product if,Inner Products,F(u, v) = ut • v,In Matlab: ut • v = u’ * v or dot (u, v),Example 2 : V = M2x3(R) = set of all 2-by-3 real matrices.,Is it an inner product?,Example 1 : V = Rn .,2.,1.,3.,4.,5.,But,Thus F(A, B) may not be an inner product. To show that it is not an inner product, we need to find A, B, C,  M23 (R) , such that F(A + B, C) F(A, C) + F(B, C).,Can you find them by Matlab ?,In Matlab:,F(A, B) = max max (abs (A’* B)),,Can you think of another inner product for M2×3 (R) ?,
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