(4.2) and (4.3), it follows that an identity matrix has a condition number equal to unity since all its eigenvalues are also equal to unity. в€’ When adopting this approach, there is a risk of selecting outliers if the data were not inspected a priori and potential outliers removed. + Using selection algorithm based on the вЂyвЂ™ reference value is advantageous to cover the wider range of вЂyвЂ™ values.

+ From the example given above, calibration data set selection based on optimality criteria improved the quality of PLS model predictions by improving the representativeness of the calibration data. This is mostly the case for data when standards cannot be prepared, for example, natural products, reaction kinetics, biological synthesis, and phenomena where the kinetics are too fast to collect samples or where, for safety reasons, it is impossible to collect lots of samples for reference measurements. Use of algorithms such as K&S and Duplex is interesting in the case of small data sets (less than 10 samples) for building and testing the model. Duplex selection applied to the y vector gives the best results, with better SEP and bias. Ranking selection has resulted in a model with better R 2 and SEP compared to random selection. However, Table 1 clearly shows the advantage of using an optimal selection algorithm compared with random selection. This comparison is based on the particular data set used in this example and of course the statistical results obtained will depend on the data set used.

Therefore, let us try to reverse the order of our factors. Still, this factoring is not quite satisfactory, since in geometric modeling the perspective transformation comes last rather than first.

Thus, we have succeeded in decomposing a singular projective transformation into simple, geometrically meaningful factors. The matrix for the perspective transformation R can be found explicitly from the eyepoint E and the plane S by the methods described in Goldman (1990), and the affine transformation A can be factored further into simple, geometrically meaningful, factors by the techniques described in Gem III.3. Thus, we have succeeded in factoring a singular projective transformation M into the product of a perspective transformation R and an affine transformation A. Therefore, the perspective projection R maps P * to Q *. The last equality holds because P lies on the line joining Q and E. The eigenvalues will be computed iteratively while the eigenvectors are obtained in the process of this iteration or by actually solving the homogeneous equation noniteratively or iteratively. The eigenvalue computation is identical with the problem of computing the roots of a polynomial equation while the eigenvector computation is identical with that of computing a solution vector x of the linear homogeneous equation (A в€’ О»I) x = 0 (null column vector), where I is the n Г— n unit matrix and the eigenvalue О» is known/computed. If we find a scalar О» and an n Г— 1 vector x so that A x = О» x then О» is one of n possible eigenvalues and x is the corresponding eigenvector of A. If A is nonsquare then we may append appropriate number of zero rows or zero columns to make it square before we talk about its eigenvalues and eigenvectors. The eigenvalues and eigenvectors are defined for an n Г— n(singular or nonsingular) matrix A and not for an m Г— n rectangular matrix, where m в‰ n. Sen, in Mathematics in Science and Engineering, 2005 5.5.8 Eigenvalues and eigenvectors of a matrix

Singular matrix