By H. Ronald Miller
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Additional info for Optimization: Foundations and Applications
However, in light of our understanding of the general form of the inverse and of the general definition of a determinant, it is easy to see where this rule comes from. 25), so that expression can be condensed to 1 ΙΑ I - | A "I which is just Cramer's rule. Here is an illustration. 6), we had A = 5 7 1 2 and Β 2000 460 26 MATRIX ALGEBRA SO 2000 7 460 2 and A fl2 - 5 2000 1 460 Using Cramer's rule *2 = 2000 7 5 7 460 2 1 2 5 2000 5 7 1 460 1 2 = 7 =f f = 260 = 100 which is the same as our earlier results.
A vector is stretched or shrunk depending on whether the absolute value of the scalar is greater or less than 1; the orientation in space is reversed or not depending on whether the sign of the scalar is negative or positive. 12 " T h e fact that multiplication of a vector by an ordinary number changes only the scale of the vector constitutes the origin of the term scalar for an ordinary number in matrix algebra. Proofs of the changes in 9 that accompany multiplication by a negative number rely on some fundamental identities from trigonometry that need not concern us here.
21 «22. Xi) "«11 fll2 _«21 «22. ^ 2 . 34) is that it represents a quadratic form of any number of variables; nothing changes except the dimensions of the matrices involved. «31 «32 «33. and A is the 3 χ 3 symmetric matrix i. X «13 where for all off-diagonal elements, a = a . An additional representation is i} }i 3 3 β(*1>*2>* ) = Σ Σ «ί/*/*/ 3 This makes clear the quadratic nature of the function. When ί =j, a variable is squared; otherwise (when / Φ j) pairs of variables are multiplied together.