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By c rao

This quantity goals to provide the elemental strategies, measurements, and methods of infrared spectroscopy and to survey, as thoroughly as attainable, its chemical functions.

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The rank of three-way and multi-way arrays is considerably more complicated than for the two-way arrays and is treated separately. Definitions and properties Suppose that a matrix A (I × J ) is given. The matrix A has J columns. 1 The matrix A defines a linear transformation from R J to R I . The range or column space of A is defined as (A) = {y|Ax = y; ∀x ∈ R J } or, stated in words, the range of A consists of all the ys for which there is an x such that Ax = y. Clearly, (A) ⊆ R I . The rank of A is the dimension of (A) and this equals the column-rank of A.

T R and p1 , . . , p R is not unique: there is rotational freedom and intensity (or scale)/signindeterminacy. In Chapter 3, some more aspects of this are outlined. 16) shows that each matrix of rank R can be written as a summation of R rank one matrices. Some important equalities follow. 17) (AA ) (v) r(A) = r(S), S being the singular values of A arranged in a diagonal matrix (see Chapter 3) Proofs of these equalities can be found in standard matrix algebra books [Magnus & Neudecker 1988, Schott 1997, Searle 1982].

A line L is sought that describes the original data as well as possible. That is, the orthogonally projected data (open circles) are as close as possible to the original data (closed circles). This is the Pearson approach: a line of closest fit [Eckart & Young 1936, Pearson 1901]. The emphasis is on the variance explained by the principal components. On the other hand, line L represents a new variable (a linear combination of the manifest variables x1 and x2 ) and the scores on this new variables are the open circles.

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