Question

3. A Mercator matrix is a matrix A E Matn,n(R) such that pA(x) = (x-aj (x-4) for some α' E R, which are not necessarily distinct, such that 0 < α' < 2. Let Mn(R) be the set of Mercator matrices. As in Tutorial 1, if A EM(R) define the logarithm of A to be the matrix given by the power series: k>0 If A, B EM,(R) then, by an argument similar to that used in Question 12(a) of Tutorial 1 Log(AB) = Log(A) + Log(B), for all A, B E Mn(R) such that AB = BA. Recall that if β E R and 0 < β < 2 then the Taylor's series expansion of ln(x) at l converges so that In@) = 〉 (-1)"+1-(β-1)K k>0 You may use all of the statements above without proof.

Note that is the $\beta$-Jordan block of size m with $\beta$'s in the diagonal and 1 to the right of the diagonal.

Answer 1

the -senie 90 ド>0 with s m 0 Ktl

NoツShmce , o(@c 2ー than with enriat mula . anib value 9 ーラ。 Omatux in