Math 217 Winter 2009 - Group Homework 9
1. The Cayley-Hamilton Theorem states that if Ais a square matrix and
p(t) is the characteristic polynomial of A:
p(t) = det(A−t·I)
then p(A) = 0. That is, if you evaluate pat A(using matrix matrix
multiplication as so on) you end up with the zero matrix. Check this
directly for the matrix
A=2 1
2 3
2. Let Abe an n×nmatrix and let {~x1, . . . , ~xn}be a basis of Rn. Show
that the matrix of the linear transformation associated to Awith re-
spect to this basis is upper-triangular if and only if
A~xk∈span{~x1, . . . , ~xk}for k= 1,2, . . . , n
3. Let Abe an n×nmatrix and suppose that the characteristic polynomial
of Ais p(t)=(λ−t)nfor some fixed λ∈R. Show that Ais similar to an
upper-triangular matrix. Do this by finding a basis with the property
in problem 2 above. You may find problem 1 helpful.
4. Let Abe a matrix such that all roots of the characteristic polynomial
of Aare real. Show that Ais similar to an upper-triangular matrix.