By Jeremy D Finn

ISBN-10: 003083239X

ISBN-13: 9780030832390

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**Example text**

_ . . _.... ,...... Ya'a. -·~·For two dimensions, fi~st cons(der A diagonal. \ JJQ ~ 1'"-0 ...... 1. of th~. 5. Th1s result can also be obtained directly by multiplying the diagonal elements of A. 1 tThe vertical lines are used in two ways. When the enclosed array is a matrix, such as IAI, they denote the determinant. When the array is a vector, such as Ivi, they denote its length. In the case of a 1 x 1 matrix or a 1-element vector, the two are equal. The Algebra of Matrices 33 In the nondiagonal case, the angle between a1 and a 2 is of consequence.

And·w arealso orthogonal vectors, but vJ. is the residual v-vector from w. , although both pairs are at right angles on the same dimension graph. Vector products can produce results that are usually represented in scalar algebra. The reader may wish to demonstrate for himself that the product of the 1 xn unit vector 1' and a conformable column vector y is simply the sum of the elements of y. The square length of ann-element unit vector is n. z) n- Since y'y = L1Y1 2 , this is the common computational form for the sample variance, but derived through vector operations.

1 0 0 4· T= 6 7 3 0 6 -2 1 0 3 (Zero)] 6 -2 1 0 4 22 Introduction T is a lower triangular matrix, with nonzero elements on or below the diagonal. p$F'·triangular ma(rix has nonzero elements on or above the diagonal. Jatrix ils square with zeros in all positions except the principal diagonal; for example, J D = [1 ~ 1~ ~l 0 0 11 has diagonal elements [ du] # 0, and off-diagonal elements [ d;;] = 0 (i # j). For simplicity, a diagonal matrix may be written in terms of only the diagonal elements; for example, D = diag (16, 15, 11) or D = diag (du, d22• ...

### A General Model for Multivariate Analysis (International Series in Decision Processes) by Jeremy D Finn

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