where ah and ai are the means of the row and column corresponding to element ahi of matrix A, respectively, whereas a is the mean of all a^'s in the matrix. This centring has the effect of positioning the origin of the new system of axes at the centroid of the scatter of objects, without altering the distances among objects. Since the sums of the rows and columns of A1 is null, A1 has at least one null eigenvalue.

Euclidean In the particular case of distances computed using the Euclidean distance distance coefficient (Dh eq. 7.34), it is possible to obtain the Gower-centred matrix A1 directly, i.e. without calculating a matrix D of Euclidean distances and going through eqs. 9.20 and 9.1, because A1 = YY' (for Y centred). This may be verified using numerical examples. In this particular case, A1 is a positive semidefinite matrix (Table 2.2).

• The eigenvalues and eigenvectors are computed and the latter are scaled to lengths equal to the square roots of the respective eigenvalues:

Due to the centring, matrix A1 always has at least one zero eigenvalue. The reason is that at most (n - 1) real axes are necessary for representing n points in Euclidean space. There may be more than one zero eigenvalue in cases where the distance matrix Degenerate is degenerate, i.e. if the objects can be represented in fewer than (n - 1) dimensions. In D matrix practice, there are c positive eigenvalues and c real axes forming the Euclidean representation of the data, the general rule being that c < n - 1.

With the Euclidean distance (D1), when there are more objects than descriptors (n > p), the maximum value of c is p; when n < p, then c < n - 1. Take as example a set of three objects or more, and two descriptors (n > p). The objects, as many as they are, may be represented in a two-dimensional space — for example, the scatter diagram of the two descriptors. Consider now the situation where there are two objects and two descriptors (n < p); the two objects only require one dimension for representation.

• After scaling, if the eigenvectors are written as columns (e.g. Table 9.7), the rows of the resulting table are the coordinates of the objects in the space of principal coordinates, without any further transformation. Plotting the points on, say, the first two principal coordinates (or more) produces a reduced-space ordination diagram of the objects in two (or more) dimensions.

Table 9.7 Principal coordinates of the objects (rows) are obtained by scaling the eigenvectors to JX .

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