Extracting Experimental Information from Large Matrices. 2. Model-Free Resolution of Absorbance Matrices: M 3 Ga ´ bor Peintler,* ,² Istva ´ n Nagypa ´ l, ² Irving R. Epstein,* ,‡ and Kenneth Kustin ‡ Institute of Physical Chemistry, UniVersity of Szeged, H-6701 Szeged, P.O. Box 105, Hungary, and Department of Chemistry, Brandeis UniVersity, MS 015, Waltham, Massachusetts 02454-9110 ReceiVed: NoVember 5, 2001 We present a new method for the decomposition of an experimental absorbance matrix into concentration and molar absorption coefficient matrices. The decomposition in general is not unambiguous; therefore, the method may yield only ranges for these matrices. The experimental matrix is not changed, so deviations from the original data can be monitored element by element. Consequently, all chemical constraints (such as stoichiometry) can be taken into account. The method, which does not require an explicit chemical model, is used to analyze the reaction between a Co(II)-EDTA complex and H 2 O 2 . I. Introduction Advanced spectroscopic instruments routinely produce large experimental data sets in matrix form. A new algorithm for handling these matrices and for determining the number of independent absorbing species (NIAS) was reported in part 1. 1 Once NIAS is known, the next step is usually the decomposition of the experimental matrix into concentration and molar absorbance matrices, based on the Beer-Lambert law (or its analogues): where A is the experimental matrix, C is the concentration matrix, and E is the matrix of molar absorption coefficients. The dimensions of these matrices are given in parentheses, where n, p, and q are the numbers of absorbing species, samples, and wavelengths, respectively. Methods used to perform the decomposition fall into either of two classes, which differ in principle. Model-based ap- proaches posit a mathematical model that describes the relation among the concentrations or between the concentrations and their time derivatives. These methods most frequently use a least squares technique 2-4 to calculate the parameters of the model, e.g., formation constants in equilibrium studies or rate constants in kinetics. Finding an appropriate model in complex kinetic or equilibrium systems requires intuition, chemical instinct, experience, and sometimes a bit of luck. In these calculations, instead of the p × n individual concentration data, one calculates only a few chemical parameters that describe the relations among them. The second category of methods, to which the approach developed here belongs, is model-free. If a model-free solution for the concentrations is known, then further evaluation, i.e., setting up an appropriate chemical model, will be much easier than using any model-based method alone. Model-free decom- position is typically based on factor analysis (FA) and its offshoots, 5 mainly principal component analysis (PCA). In part 1, we detailed obstacles to the evaluation of large experimental matrices in connection with matrix rank analysis (MRA). These problems, particularly interdependence of the primary data, must be considered when FA is used as well. Additional issues should be taken into account when applying such techniques. The most important of these problems is that the solution of eq 1 is rarely, if ever, unique. One trivial example is that if a C-E matrix pair is a solution for eq 1 then the R × C-E/R matrix pair also satisfies eq 1 where R is any positive constant. This difficulty can be remedied by introducing additional, often chemically evident, constraints (e.g., the sum of the concentrations of absorbing species is constant). However, the application of stoichiometric constraints is hardly ever sufficient to yield a unique solution, which can be calculated only if the measured matrix contains independent experimental information for every species. A unique solution cannot be extracted if the elements of the matrix A (i.e., the measured absorbances) are determined by more than one absorbing species. Almost all variants of FA are based on an eigenvalue calculation for the matrix AA ) A T × A. We have shown in part 1 that this approach may be misleading, because the statistical criteria of the eigenvalue calculation are not valid for large matrices originating from spectroscopic data acquisition systems. The nonnegativity of the elements of the absorbance matrix is an important constraint in most experimental methods (e.g., UV-vis spectrophotometry). A negative element may only be accepted if its absolute value is smaller than the experimental error. The aim of the present work is to introduce a new algorithm to overcome the difficulties outlined above. The new method is called M 3 (model-free modeling with matrices). As we shall see, the method is capable of determining NIAS and calculating a large number of possible C and E matrix pairs. The algorithm is illustrated on a real example, the reaction of a cobalt(II)- EDTA (EDTA ) ethylenediaminetetraacetate) complex with hydrogen peroxide. II. The M 3 Algorithm The goal of M 3 is to minimize the target function * To whom correspondence should be addressed. ² University of Szeged. ‡ Brandeis University. Sn (C(n), E(n)) ) ∑ i)1 p ∑ j)1 q |A ij - ∑ k)1 n c ik ǫ kj | (2) A(p × q) ) C(p × n) × E(n × q) (1) 3899 J. Phys. Chem. A 2002, 106, 3899-3904 10.1021/jp014064n CCC: $22.00 © 2002 American Chemical Society Published on Web 03/22/2002