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Extra info for Handbook of Chemometrics and Qualimetrics Part B
6. Schematic illustration of four types of special matrix products: the matrix-by-vector product, the vector-by-matrix product, the outer product and the scalar product between vectors, respectively from top to bottom. 31) xxy = z (read as x cross y) where x, y and z have the same dimension n. Geometrically, we can regard x and y as two vectors drawn from the origin ofS"^ and forming an angle (x, y) between them. 33) x'y = z For example: •-1 [3 0 2 -1] 4 -2 0 :3x(-l) + 0 x 4 + 2 x ( - 2 ) + (-l)x0 = - 7 26 The product of a matrix with a diagonal matrix is used to multiply the rows or the columns of a matrix with given constants.
Geometrically, this point corresponds with the center of mass of the row-pattern P'^ of points formed in column-space S^. In this chapter we assume that all points in a pattern are given the same mass. In data analysis it is possible to assign different masses to individual 43 points, thus giving more weight to certain points than to others. This aspect will be further developed in Chapter 32. 60) P P j where 1^ is the sum vector of dimension p. The elements of m^ represent the coordinates of the row-centroid, which corresponds with the center of mass of the column-pattern P^ of points formed in row-space S"^.
For example, by working out the rows-by-column product one can 21 easily verify that: "l 0' "1 0 0] [3 4' '3 4' 0 1 — 0 1 0 2 -1 — 2 -1 0 0 ij [o 3 0 3_ "3 4 2 -1 0 3 A matrix is orthogonal if the product with its own transpose produces a diagonal matrix. 21) where D„ is a diagonal matrix of dimension n and D^ is a diagonal matrix of dimension p. 2. In the former case we state that X is row-orthogonal, while in the latter case X is said to be columnorthogonal. 22) where I^ and I^ have been defined before.