matlab讲义4.pptVIP

Determinant Inner 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 ratio 0.1090 --- 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) ?