4.3.Asymptotic Spectral Distributions for Hermitian matrix Now,from the Cayley-Hamilton theorem,any matrix is a (matrix- valued)root of its characteristic polynomial.That is,denoting P(x)the characteristic polynomial HHH+o21,i.e. P(x)=det(HH+2I -xI) it is clear that P(HHH+021)=0.Since the determinant above can be written as a polynomial of x of maximum degree n,P(x)expresses as P(x)=∑a,x i=0 for some coefficients ao,...,an to determine. 11
11 4.3. Asymptotic Spectral Distributions for Hermitian Matrix Now, from the Cayley-Hamilton theorem, any matrix is a (matrixvalued) root of its characteristic polynomial. That is, denoting P(x) the characteristic polynomial HHH+σ2 In , i.e. it is clear that P(HHH+σ2 In ) = 0. Since the determinant above can be written as a polynomial of x of maximum degree n, P(x) expresses as for some coefficients a0 ,…,an to determine. H 2 det P x x H H I I n n 0 n i i i P x a x
4.3.Asymptotic Spectral Distributions for Hermitian Matrix we then have 0=P(H+oL,）=∑a(HH+a1)' from which -a,=2a,(HH+o1,》 i= Multiplying both sides by (HHH+021)1,this becomes (H+,)厂'-∑g(HH+o月 8(u 12
12 4.3. Asymptotic Spectral Distributions for Hermitian Matrix we then have from which Multiplying both sides by (HHH+σ2 In ) -1 , this becomes H 2 H 2 0 0 n i n i n i P a H H I H H I H 2 0 1 n i i n i a a H H I 1 1 H 2 H 2 1 0 1 1 2( 1) H 1 0 0 1 n i i n n i n i i i i j i j a a a i a j H H I H H I H H
4.3.Asymptotic Spectral Distributions for Hermitian Matrix and therefore=HHy can be rewritten under the form =∑6，(HHHy 8台2少… Obviously,the effort required to compute(HHH+o21)is equivalent to the effort required to compute the above sum. 13
13 4.3. Asymptotic Spectral Distributions for Hermitian Matrix and therefore can be rewritten under the form With . Obviously, the effort required to compute (HHH+σ2 In ) -1 is equivalent to the effort required to compute the above sum. 1 H H 0 ˆ n i i i b x H H H y 1 2( 1) 0 0 1 i i i j i j a i b a j 1 H 2 H ˆ n x H Η I H y