Mathematical Methods for Physics and Engineering - Matematica.NET

8.18 SIMULTANEOUS LINEAR EQUATIONSWe already know from the above discussion, however, that the non-zero eigenvalues ofthis matrix are equal to those of AA † found above, and that the remaining eigenvalue iszero. The corresponding normalised eigenvectors are easily found:λ 1 =16 ⇒ v 1 = 1 (1 1 1 1)T2andsothematrixV is given byλ 2 =9 ⇒ v 2 = 1 (1 1 − 1 − 1)T2λ 3 =4 ⇒ v 3 = 1 (−1 1 1 − 1)T2λ 4 =0 ⇒ v 4 = 1 (1 − 1 1 − 1)T2⎛V = 1 ⎜⎝2⎞1 1 −1 11 1 1 −1 ⎟1 −1 1 1⎠ . (8.140)1 −1 −1 −1Alternatively, we could have found the first three columns of V by using the relation(8.135) to obtainv i = 1 A † u i for i =1, 2, 3.s iThe fourth eigenvector could then be found using the Gram–Schmidt orthogonalisationprocedure. We note that if there were more than one eigenvector corresponding to a zeroeigenvalue then we would need to use this procedure to orthogonalise these eigenvectorsbefore constructing the matrix V.Collecting our results together, we find the SVD of the matrix A:⎛1 0 0A = USV † ⎜= ⎝ 03045− 4 5 535⎛⎞ ⎛⎟⎠ ⎝ 4 0 0 0⎞0 3 0 0 ⎠⎜0 0 2 0 ⎝this can be verified by direct multiplication. ◭1212− 1 212112122− 1 2− 1 21212− 1 212− 1 212− 1 2Let us now consider the use of SVD in solving a set of M simultaneous linearequations in N unknowns, which we write again as Ax = b. Firstly, considerthe solution of a homogeneous set of equations, for which b = 0. As mentionedpreviously, if A is square and non-singular (and so possesses no zero singularvalues) then the equations have the unique trivial solution x = 0. Otherwise,anyof the vectors v i , i = r +1,r+2,...,N, or any linear combination of them, willbe a solution.In the inhomogeneous case, where b is not a zero vector, the set of equationswill possess solutions if b lies in the range of A. To investigate these solutions, itis convenient to introduce the N × M matrix S, which is constructed by takingthe transpose of S in (8.131) and replacing each non-zero singular value s i onthe diagonal by 1/s i . It is clear that, with this construction, SS is an M × Mdiagonal matrix with diagonal entries that equal unity for those values of j forwhich s j ≠ 0, and zero otherwise.Now consider the vectorˆx = VSU † b. (8.141)305⎞⎟⎠ ;