NMR Water Artifact Removal - Biomedical Signal Analysis: Contemporary Methods and Applications - page 385

14.3 Computing the Eigendecomposition of Symmetric Pencils

The matrix pencil ( R x,1 , R x,2 ) of zero mean data comprises two corre-
lation matrices of the data. The first matrix is computed as follows:

1
N S ( ω 2 ,t 1 ) S H ( ω 2 ,t 1 ) ,

R x,1 = (14.10)

with N = 2048 representing the number of samples in the ω 2 domain
and S H the conjugate transpose of the matrix S . The second correlation
matrix R x,2 of the pencil has been computed after filtering each single
spectrum (each row of S ( ω 2 ,t 1 )) with a bandpass filter of Gaussian
shape centered in the spectrum and having a variance in the range of
1

σ 2 ≤

≤ 4. Both matrices of the pencil are of dimension 128 × 128,

since we assume as many sources as there are sensor signals.
A very common approach to computing the eigenvalues and eigen-
vectors of a matrix pencil is to reduce the GEVD statement

R x2 E = R x1 EΛ

to the standard eigenvalue decomposition (EVD) problem, which is of
the form

CZ = ZΛ .

The strategy that we will follow is first to solve the eigendecomposi-
tion of the matrix R x1 , giving

R x1 = SDS T = S 1/2 D 1/2 S T SD 1/2 S T

= WW .

Substituting this result into the GEVD statement and defining Z = WE
yields the transformed equation

W −1 R x2 W −1 Z = ZΛ ,

which is the standard EVD form of a real symmetric matrix C =
W −1 R x2 W −1 if the matrix R x2 is also symmetric positive definite and
the transformation matrix W −1 is obtained as

W −1 = SD −1/2 S T .

While the eigenvalues of the matrix pencil are available from the solution
of the EVD of the matrix C , the corresponding eigenvectors are obtained
via E = W −1 Z .