ATLAB Lecture 4 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself, as it does for an ordinary differential equation. On the other hand. the singular value decomposition is the appropriate tool for analyzing a mapping from one vector space into another vector space, possibly with a different dimension. Most systems of simultaneous linear equations fall into this second category If A is square, symmetric, and positive definite, then its eigenvalue and singular value decompositions are the same. But, as A departs from symmetry and positive definiteness, the difference between the two decompositions increases. In particular, the singular value decomposition of a real matrix is always real, but the eigenvalue decomposition of a real nonsymmetric matrix might be complex >>A=9 7 >>US, V=svd(A) %the full singular value decomposition -0.61050.71740.3355 -0.66460.2336-0.7098 -0.43080.656306194 14.9359 05.1883 -0.69250.7214 -0.7214-0.6925 You can verify that U Sy is equal to a to within roundoff error. >>U,S, V=svd(A, 0) %the economy size decomposition is only slightly smaller -0.61050.7174 -0.6646-0.2336 -04308-0.6563 14.9359 05.1883 -0.69250.7214 -0.7214-0.6925 Again, USW is equal to a to within roundoff error ☆ MATLAB >>A=round(10*randn(3) Lec4-4MATLAB Lecture 4  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec4­4  The eigenvalue decomposition is the appropriate tool for analyzing a matrix  when it represents a mapping from a vector space into itself,  as it does for an ordinary differential  equation.  On the other hand,  the singular value decomposition is the appropriate tool  for analyzing a mapping from one vector space into another vector space,  possibly with a different  dimension.  Most systems  of simultaneous  linear equations  fall into this  second  category.  If A is square,  symmetric,  and positive definite,  then  its  eigenvalue and singular value  decompositions are the same. But, as A departs from symmetry and positive definiteness, the  difference between the two  decompositions  increases.  In particular,  the singular value decomposition of a real matrix is always real,  but the eigenvalue decomposition of a real,  nonsymmetric matrix might be complex.  >> A =[9  4  6  8  2  7];  >> [U,S,V] = svd(A) %the full singular value decomposition  U =  ­0.6105  0.7174  0.3355  ­0.6646  ­0.2336  ­0.7098  ­0.4308  ­0.6563  0.6194  S =  14.9359  0  0  5.1883  0  0  V =  ­0.6925  0.7214  ­0.7214  ­0.6925  You can verify that U*S*V' is equal to A to within roundoff error.  >> [U,S,V] = svd(A,0) %the economy size decomposition is only slightly smaller.  U =  ­0.6105  0.7174  ­0.6646  ­0.2336  ­0.4308  ­0.6563  S =  14.9359  0  0  5.1883  V =  ­0.6925  0.7214  ­0.7214  ­0.6925  Again, U*S*V' is equal to A to within roundoff error.  ² MATLAB >> A = round(10*randn(3))