A General Model for Multivariate Analysis (International by Jeremy D Finn

By Jeremy D Finn

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1 0 0 OJ [00 01 01 00 0 0 1 0 The s~scrir;>Lon Us Y_s~d to denote its row and column order. A[ symmetric matrixh any square array in which the elements above and -J below the-principal diagonal, element for element, are identical. That is, the first row is identical to the first column, the second row is identical to the second column, and so on. For example, s= 13 [ 102 -6 4 4] 102 -6 12 17 -4 17 0 0 -4 0 -8 Each element [s;;] is identical to corresponding element [s;;]. For simplicity, only the lower symmetric half of a symmetric matrix is written explicitly.

Since the maximum value of x' Ax can become infinite, xis frequently restricted to having unit length; that is, x'x = 1. Let us introduce A. 1) The maximum value A. I)x = 0 The maximum A. and corresponding vector x are the non-null solutions of these equations. I is an identity matrix of order n. I and solve for x by premultiplying both sides of Eq. I)- 1 The only solution for x would then be x= 0. I is singular and cannot be inverted. rLA of Eq. 6 .. 'LS the a5s0ciated charagteristl9. 3 we· --- ---~-~-·-~"'".

_ . . _.... ,...... 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.

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