Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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8.18 SIMULTANEOUS LINEAR EQUATIONS


We already know from the above discussion, however, that the non-zero eigenvalues of
this matrix areequalto those ofAA†found above, and that the remaining eigenvalue is
zero. The corresponding normalised eigenvectors are easily found:


λ 1 =16 ⇒ v^1 =^12 (1 1 1 1)T
λ 2 =9 ⇒ v^2 =^12 (1 1 − 1 −1)T
λ 3 =4 ⇒ v^3 =^12 (− 111 −1)T
λ 4 =0 ⇒ v^4 =^12 (1 − 11 −1)T

andsothematrixVis given by


V=


1


2





11 − 11


11 1− 1


1 −11 1


1 − 1 − 1 − 1




. (8.140)


Alternatively, we could have found the first three columns ofVby using the relation
(8.135) to obtain


vi=

1


si

A†ui fori=1, 2 , 3.

The fourth eigenvector could then be found using the Gram–Schmidt orthogonalisation
procedure. We note that if there were more than one eigenvector corresponding to a zero
eigenvalue then we would need to use this procedure to orthogonalise these eigenvectors
before constructing the matrixV.
Collecting our results together, we find the SVD of the matrixA:


A=USV†=





10 0


0 35 −^45


(^04535)







4000


0300


0020










1
2

1
2

1
2

1
2
1
2

1
2 −

1
2 −

1
2
−^121212 −^12
1
2 −

1
2

1
2 −

1
2







;


this can be verified by direct multiplication.


Let us now consider the use of SVD in solving a set ofMsimultaneous linear

equations inNunknowns, which we write again asAx=b. Firstly, consider


the solution of a homogeneous set of equations, for whichb= 0. As mentioned


previously, ifAis square and non-singular (and so possesses no zero singular


values) then the equations have the unique trivial solutionx= 0 .Otherwise,any


of the vectorsvi,i=r+1,r+2,...,N, or any linear combination of them, will


be a solution.


In the inhomogeneous case, wherebis not a zero vector, the set of equations

will possess solutions ifblies in the range ofA. To investigate these solutions, it


is convenient to introduce theN×MmatrixS, which is constructed by taking


the transpose ofSin (8.131) and replacing each non-zero singular valuesion


the diagonal by 1/si. It is clear that, with this construction,SSis anM×M


diagonal matrix with diagonal entries that equal unity for those values ofjfor


whichsj= 0, and zero otherwise.


Now consider the vector

xˆ=VSU†b. (8.141)
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