By Doz. Dr. sc. Georg Heinig, Dr. rer. nat. Karla Rost (auth.)
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Extra info for Algebraic Methods for Toeplitz-like Matrices and Operators
D. )d(A) 1 a(A) • a 0 (A)d(A)o Clearly, d(A) ia a polynomial with real coefficients. Applying Prop. 14 we obtain B 0 ,a0 ) = Res(d,a0 ) T [-iBez(a 0 OJ 0 Res(d,a0 ). 5 1 Res(d,a0 ) is regular. Furthermore, Res(d,a0 )T = Res(d,a0 )• and therefore, Sylvester's inertia theorem can be applied. This yields p~(B) = p±(B0 ), p 0 (B) = p 0 (B0 ) + r, •= -i Bez(a0 ,i0 ) and r is the degree of d(A)o By Prop. 9 B0 is regular, hence, p 0 (B) = r. On the other hand, we have r = ~ 0 (a) + 2Q(a), and the last part of the theorem is proved.
Now we are going to apply Prop. 15 to the particular case b(A) = a(A), where a(A) is assumed to be monic. For short we shall write B J= -1 Bez(a,a). We restrict ourselves to the situation that a(A) 49 has no roots distributed symmetric relative to the real line. (As remarked. 7 the general oase can be led to this one). , o(A) the imaginary part of a(A), and let e E IR be chosen such that the (real) polynomial b(A) + eo(A) has simple roots A1 , ••• ,An only (we have already mentioned that such e A~+ 1 , ••• , ~+1 ,A~+2 exists).
2, £• In this subsection we deal merely with the case r = R, G+ is the upper, G_ is the lower half-plane, and we shall omit the subindex r. Given an Hermitian matrix U the number of positive, negative and zero eigenvalues of U (counting multiplicities) will be denoted by p+(u), p_(u), p 0 (U), respectively. The triple In U I= (p+(U), p_(U), p 0 (U)) is called inertia Qf rank u = p+(u) ~ Hermitian matrix ~· Clearly, + p_(u), sgn U • p+(u) - p_(u), where sgn U denotes the signature of U, The main result of this subsection is the theorem of Hermite.
Algebraic Methods for Toeplitz-like Matrices and Operators by Doz. Dr. sc. Georg Heinig, Dr. rer. nat. Karla Rost (auth.)