MA554_AUG09 - Math 554 Qualifying Exam August 2009(Jiu-Kang...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 554 Qualifying Exam. August, 2009 (Jiu-Kang Yu) 1. Let J e be the e × e complex matrix with J j +1 ,j = 1 for j = 1 , . . . , e - 1, J i,j = 0 if i 6 = j + 1. It is the so-called e × e nilpotent Jordan block. Let e 1 ≥ · · · ≥ e r be a decreasing sequence of positive integers and let A = J e 1 ,...,e r be the direct sum of J e 1 , . . . , J e r . (a) (8 points) Compute dim ker A m , for m 0. (b) (8 points) Show, without using the structure theorem, that if J e 1 ,...,e r is similar to J f 1 ,...,f s (where f 1 ≥ · · · ≥ f s is another decreasing sequence of positive integers), then r = s and e i = f i for i = 1 , . . . , r . (c) (8 points) What is the Jordan form of A 2 ? It is enough to describe the sizes (and eigenvalues) of its Jordan blocks. (d) (8 points) What is the Jordan form of A 2 + A ? 2. (10 points) Let F 2 be the finite field Z / 2 Z . How many similarity classes of 3 × 3 invertible matrices over F 2 are there? You may use the fact that there are 2,1,2 monic irreducible polynomial of degree 1 , 2 , 3 respectively. It may help to consider the rational canonical forms.
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern