Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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)