Systems Biology (Methods in Molecular Biology)

A.7 Singular Value
Decomposition

EveryA, matrixmn,mncan be decomposed as

A¼UΣVT ð 39 Þ

where (.)Tdenotes the transposed matrix andUismnmatrix,

Visnnmatrix satisfying

UTU¼VTV¼VVT¼In ð 40 Þ

andΣ¼E½σ 1 ,...,σnŠa diagonal matrix.

Theseσi’s,σ 1 /gσ 2 ,...,σn0 are the square root of the

nonnegative eigenvalues ofATAand are called as the singular values

of matrix A. As it is well known from linear algebra, see i.e., Press

et al. [29] singular value decomposition is a technique to compute

pseudoinverse for singular or ill-conditioned matrix of linear sys-

tems. In addition this method provides least square solution for

overdetermined system and minimal norm solution in case of

undetermined system.

The pseudoinverse of a matrixA,mnis a matrixA+,n∗m

satisfying

AAþA¼A,AþAAþ¼Aþ,ðAþAÞ∗¼AþA,ðAAþÞ∗

¼AAþ ð 41 Þ

where (.)∗denotes the conjugate transpose of the matrix.

Always exists a uniqueA+which can be computed using SVD:

1. Ifm>¼nandA¼UΣVT, then

Aþ¼VΣ^1 UT ð 42 Þ

whereΣ^1 ¼E½ 1 =σ 1 ,...,1=σnŠ

2. Ifm<n, then compute the (AT)+, pseudoinverse ofATand
   then

Aþ¼ððATÞþÞT ð 43 Þ

A.8 Newton-Raphson
Method
with Pseudoinverse

The idea of using pseudoinverse in order to generalize of Newton

method is not new but has been suggested by different authors,

among others we may cite Haselgrove [30]. It means that in the

iteration formula, the pseudoinverse of the Jacobian matrix will be

employed,

xiþ 1 ¼xiJþðxjÞfðxiÞð 44 Þ

90 Rodolfo Guzzi et al.