Advanced Calculus fi..

Chapter 1 Vectors and Matrices 41The product of the last two factors is an n x 1 column vector whose ith entry isailxl + . . . + a,,x,. The product of the 1 x n row vector (XI, . . . , x,) and this n x 1column vector is a 1 x I matrix-that is, a number; in fact, it is precisely the numberon the right-hand side of (1.84). Therefore (1.85) is indeed another way of writing(134).As an illustration, we write the quadratic form Q(x) of (1.83) as follows:When expanded, this becomesas expected."1.13 ORTHOGONAL MATRICESLet A be a real n x n matrix. Then A is said to be orthogonal ifAA' = I. (1.86)Hence, A is orthogonal if and only if A-' = A', that is, if and only if the inverseof A equals the transpose of A. Thus every orthogonal matrix is nonsingular. Thefollowing are examples of orthogonal matrices:Let us consider the row vectors ul , u2 of A as vectors in the xy-plane:Then we observe that ul and u2 are both unit vectors and that ul . u2 = 0, so thatul, u2 are perpendicular. A similar statement applies to the column vectors of A:We can proceed similarly with the row vectors or column vectors of B, regardingthem as vectors in space: ul = fi + $j + ik, . . . . Again we verify that the rowvectors, or the column vectors, are mutually perpendicular unit vectors.