Global analysis of transient optical spectra

The r rows of the matrix, i = 1,. .., r, represent the absorption spectra of the sample at a given time tt and the c columns, j = 1. c, collect the absorbances at a given wavenumber 

Matrix operations are best performed by computer only some definitions, but not the numerical procedures, are given here. Matrices of equal size (r x c) can be added (or subtracted) by adding (or subtracting) their corresponding elements, A B = (Ay By). The product C=A B of two matrices can be formed only if the number of columns of A equals the number of rows of B. The elements Q/c of C are then given by 

A square matrix is singular if its determinant IIAII (footnote h at end of Section 1.4) vanishes. The inverse A 1 of a matrix A is defined by the equation A A = E, where E is the identity matrix, E = ( 5,j) and 5,j is Kronecker s delta, which takes the value of 1 for i = j and 0 otherwise, so that E A =A E=A. The inverse of A can be determined if and only if, it is non-singular, IIAII 0. Diagonalization of a square matrix A is the equivalent of finding its eigenvalues and eigenvectors (Section 1.4). 

These matrix operations are easily programmed with mathematical software packages. In MATLAB, for example, only two statements, M = A A and [V,S] = eig(M) , are required 

The row vectors in matrix Vred define points in the three-dimensional space of the orthogonal eigenvectors. They lie in (or near, in the presence of noise) a plane, consistent 

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