Linear algebra with cubic roots

Question

Problems: ?

Linear Algebra with Cubic Roots

This problem is a partial investigation of which n×n matrices over ? have cube roots; that is, for which n × n matrices A over ? there is an n × n B over ? such that A = B3. Since ? is algebraically closed, every n × n matrix over ? is similar over ? to a matrix in Jordan canonical form.

A. Suppose that A is a nilpotent n × n matrix over ? and that there is an integer k,k ≥ n/3, such that A_k 6≠ 0. Show that A does not have a cube root. (Think about Exercise 6.3.5 of Hoffman and Kunze, for this and for part C.) Thus not all n × n matrices over ? have cube roots.

B. Show that A (do not assume nilpotent) has a cube root iff the Jordan canonical form of A has a cube root.

C.  Find c₁ ∈ Q such that if N is a nilpotent 2 × 2 matrix over ? and M = I + c₁N, then M³ = I + N. That is, M is a cube root of I + N. Find c₂ ∈ Q such that if N is a nilpotent 3 × 3 matrix over ? and M = I + c₁N + c₂N², then M³ = I + N. Again, M is a cube root of I + N.

It is not difficult to prove by induction that for any k and any nilpotent k × k matrix N over ? , there is a similar formula for a cube root of I + N. (This can also be proved using the binomial series for (1 + t)^1/3.) Assume that result for the remaining parts of this problem.

D. Show that if c is a nonzero complex number and N is a nilpotent k × k matrix over C, then cI + N has a square root. (First look at I +1/ cN.)

E. Show that if A (do not assume nilpotent) is an invertible n × n matrix over ? then A has a cube root.

*6.3.5 As referred to in parts A and C states: Let V be an n-dimensional vector space and let T be a linear operator on V. Suppose that there exists some positive integer k so that Tk = 0. Prove that T^m = 0.