Question

5. Let A € Mnxn(C) with characteristic polynomial p(x) = cxºII-1(d; – x) and li + 0, Vi, a E Z>o. Show that if dim(ker(A))+k=n, then A= C2 for some complex matrix C.

Answer 1

Solution Guven that Ae Mnxo (e) with characterih'c polynomial Þ(w = cv? ($; -x) , di totii at Zyo · (x) = x^ (2-x)(x2-x) (2x-) so, the characteristic roots of matx A are — f(x) = 0 =) ca? (1,-2) (12-2)... (k-1)=0 =) 39 = 0, 1,-= 0, 12 - x = 0 , ...,Ak-4 = 0 => x = 0,0,...,0,1,,dz, crosda O a times we know that - . no. of eigen values of A =n =) atk =n => a =n-k - 0 Also given that dim (KerA) +1=1 1. =) dim (ker A) =n-k comparing equations of ©, we have - dim (kerA) = a. Now, we know that it doo da, ... /p are eigen values of a matx B , then xirida, ..., loe are eigenvalues of B

Let there exists a mamix c with eigen values - 0,0,...,0, si, siz ... Vic, then a times complex nors, since 1,, 12, ...,dk are complexno's the eigen values of matha (² will be o, o, ...,O, atimes do, 12, ..., dk, which is mano'A. Thus, there exists a complex matrix C such that A=c² . for eg: 100. - ...01 10.0.... ..0l • O .. ...0 = 1 Oo....di...o 0 0 : : 0 A2 ..o | oo . . 100 .....TANT ló Ö.......The 0.0 oooo. - x -