X 2 xx 2 M2

is never negative and is zero if and only if all the elements of X are zero, i.e., if X=0. This product will be denoted by X2 and its positive square root by |X| or by X, if no confusion results. The scalar |X| is called the norm or magnitude of the vector, X, and can be thought of as the length of the vector. Thus, with our definition of the inner product,

which would not be true if we defined the inner product using the transpose rather than the adjoint, because the square of a complex number generally is not positive.

If we multiply an nxl row matrix (on the left) by a 1 Xm matrix (on the right), we obtain an n X m matrix. This leads to the definition of the outer product of two vectors

If the vectors are real, the adjoint of Y is the transpose of Y, and the j/th element of the outer product is Xt Yj.

Matrix division can be defined in terms of matrix inverses, which are discussed in Section C.4.

C J Trace, Determinant, and Rank

Two useful scalar quantities, the trace and the determinant, can be defined for any square matrix. The rank of a matrix is defined for any matrix.

The trace of an n X n matrix is the sum of the diagonal elements of the matrix tTA = 2 AH

The trace of a product of square matrices is unchanged by a cyclic permutation of the order of the product tr(ABC)= ¿SS A0BJkCki=tr(CAB) (C-29)

The determinant of an n X n matrix is the complex number defined by detA=\Av\ = '2(-iyAlpAw--A

where the set of numbers {p\,pi,---,p„} is a permutation, or rearrangement, of {l,2,...,/i}. Any permutation can be achieved by a sequence of pairwise interchanges. A permutation is uniquely even or odd if the number of interchanges required is even or odd, respectively. The exponent p in Eq. (C-30) is zero for even permutations and unity for odd ones. The sum is over all the n\ distinct permutations of {l,2,...,/i}. It is not difficult to show that Eq. (C-30) is equivalent to deU = 2 ("1 )'*JAyMv

for any fixed i=\,2,...,n, where My is the minor of Ay, defined as the determinant of the (n-l)X(/i- 1) matrix formed by omitting the ith row andy'th column from A. This form provides a convenient method for evaluating determinants by successive reduction to lower orders. For example,

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