Eliminating complex eigenvectors and eigenvalues in multiway analyses using the direct trilinear decomposition method

Abstract

SUMMARYThe direct trilinear decomposition method (DTDM) is an algorithm for performing quantitative curve resolution of three-dimensional data that follow the so-called trilinear model, e.g. chromatography-spectroscopy or emission-excitation fluorescence. Under certain conditions complex eigenvalues and eigenvectors emerge when the generalized eigenproblem is solved in DTDM. Previous publications never treated those cases. In this paper we show how similarity transformations can be used to eliminate the imaginary part of the complex eigenvalues and eigenvectors, thereby increasing the usefulness of DTDM in practical applications. The similarity transformation technique was first used by our laboratory to solve the similar problem in the generalized rank annihilation method (GRAM). Because unique elution profiles and spectra can be derived by using data matrices from three or more samples simultaneously, DTDM with similarity transformations is more efficient than GRAM in the case where there are many samples to be investigated.

Short Communication: On solving generalized eigenvalue problems using Matlab

“…As pointed out by Li and co-workers, 5,6 degenerate eigenvalues in the GRAM solution correspond to a rotational ambiguity which cannot be resolved. This equally holds true for the complex conjugate eigenpairs that are handled by the second similarity transformation.…”

“…Other typical examples in the context of GRAM are given by Li and co-workers. 5,6 In their examples eigenvectors corresponding to real eigenvalues have an imaginary contribution that is not negligible. From a strictly mathematical point of view this is still a valid solution to the generalized eigenvalue problem as pointed out above.…”

“…As a result additional programming is necessary in order to eliminate this complex part. Li and co-workers 5,6 propose to use two similarity transformations without, however, hinting at the origin of the problem. The first similarity transformation is devised to remove the arbitrariness due to the Euclidean normalization: the transformation matrix is a diagonal matrix and the diagonal elements have modulus unity (see Table 5 in Reference 5).…”

“…An adequate modification of Matlab's eigensolution is provided by the similarity transformations proposed by Li and co-workers within the context of the generalized rank annihilation method. 5,6 ACKNOWLEDGEMENT This work was supported by the Center for Process Analytical Chemistry (CPAC), a National Science Foundation Industry University Cooperative Research Center at the University of Washington.…”

Evaluation of variation matrix arrays by parallel factor analysis

“…If the same solution is reached several times, there is little chance for a local minimum to be found due to an unfortunate initial guess. In [10,[36][37][38] it is proposed that one use starting value based on generalized eigenvalue decomposition. Furthermore, depending on the characteristics of the studied system, constraints can be applied that results in better estimates of loadings and increasing the speed of the modeling processes [22].…”