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