This is the matrix analog of the polar form of a

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Unformatted text preview: egal to print, duplicate, or distribute this material Please report violations to [email protected] for a permutation matrix Q. Thus det (Pk ) = det FT F ≥ 0 (Exercise 6.1.10). Proof of (7.6.12) =⇒ (7.6.9). If Ak is the leading k × k principal submatrix of A, and if {µ1 , µ2 , . . . , µk } are the eigenvalues (including repetitions) of Ak , then I + Ak has eigenvalues { + µ1 , + µ2 , . . . , + µk }, so, for every > 0, det ( I + Ak ) = ( + µ1 )( + µ2 ) · · · ( + µk ) = k k−1 + s1 D E + · · · + sk−1 + sk > 0 because sj is the j th symmetric function of the µi ’s (p. 494), and, by (7.1.6), sj is the sum of the j × j principal minors of Ak , which are principal minors of A. In other words, each leading principal minor of I + A is positive, so I + A is positive definite by the results on p. 559. Consequently, for each nonzero x ∈ n×1 , we must have xT ( I + A)x > 0 for every > 0. Let → 0+ (i.e., through positive values) to conclude that xT Ax ≥ 0 for each x ∈ n×1 . T H IG R Quadratic Forms For a vector x ∈ defined by n×1 and a matrix A ∈ Y P T n n×n , the scalar function n f (x) = x Ax = aij xi xj (7.6.13) i=1 j =1 is called a quadratic form. A quadratic form is said to be positive definite whenever A is a positive definite matrix. In other words, (7.6.13) is a positive definite form if and only if f (x) > 0 for all 0 = x ∈ n×1 . O C Because xT Ax = xT (A + AT )/2 x, and because (A + AT )/2 is symmetric, the matrix of a quadratic form can always be forced to be symmetric. For this reason it is assumed that the matrix of every quadratic form is symmetric. When x ∈ C n×1 , A ∈ C n×n , and A is hermitian, the expression xH Ax is known as a complex quadratic form . Example 7.6.3 Diagonalization of a Quadratic Form. A quadratic form f (x) = xT Dx is said to be a diagonal form whenever Dn×n is a diagonal matrix, in which n case xT Dx = i=1 dii x2 (there are no cross-product terms). Every quadratic i T form x Ax can be diagonalized by maki...
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