Assignment 9
This assignment is worth fifty points. It is due Wednesday, November 22, 2006 before 3 PM. (Turn it
in to my office or my department mailbox.)
None of these problems are from chapter seven of our textbook; please do not use the theorems from
that chapter.
1. (Hoffman & Kunze, page 208: 3.)
(a) Let Fbe a family of commuting 3 ×3 complex matrices. How many linearly independent
matrices can Fcontain?
(b) What about the n×ncase?
2. Let Jnbe the n×nmatrix of all ones. (So J3:=
1 1 1
1 1 1
1 1 1
.)
(a) Find a basis of eigenvectors for Jn.
(b) Let Inbe the n×nidentity matrix and s, t scalars. Diagonalize the matrix sIn+tJn.
(c) Compute the determinant of sIn+tJn.
3. Let Lnbe the n×n(0,1)-matrix with ones on the “anti-diagonal”, that is Li,j =δi,n+1−i.
(a) Find a basis of eigenvectors for Ln.
(b) Let Mn(a, b) be a 2n×2nmatrix with aon the diagonal, bon the “anti-diagonal”, and zeros
elsewhere – that is, the (i, i) entries of Mare a, the (i, n + 1 −i) entries of Mare b, and all
the rest of the entries are zero.
(For example, here is M3(2,5) :
2 0 0 0 0 5
0 2 0 0 5 0
0 0 2 5 0 0
0 0 5 2 0 0
0 5 0 0 2 0
5 0 0 0 0 2
.)
(c) Diagonalize Mn(a, b).(Note that Mn(a, b) is in the linear algebra generated by L2n.)
4. Let Pbe the permutation matrix associated with the permutation (1,2,3,4,5).
(a) Write out P.
(b) Given a 5 ×5 matrix A, what is the effect of pre-multiplying Aby P? (That is, how does A
compare to P A?)
What is the effect of post-multiplying Aby P? (How do Aand AP compare?)
(c) Show that the minimal polynomial of Pis x5−1.
(d) Let ζ:= e2πi
5= cos( 2π
5) + isin(2π
5).Note that ζ5= 1.Show that the vector α1:=
1
ζ
ζ2
ζ3
ζ4
is an
eigenvector for P.
