Additional problems for section 25
1. Let Tbe a linear transformation on a vector space Vover the complex number
field Cwith a basis {u, v, w }, such that
T(u) = u−v
T(v) = u+ 3v
T(w) = −u−4v−w.
(a) Find the elementary divisors of T.
(b) Find the Jordan canonical form of T.
(c) Find a basis v1,v2,v3, of V, such that the matrix of Tin this basis is the
Jordan canonical form of T.
2. Continuation of Problem 8 from section 25 page 226
(a) Let Abe a matrix, whose elementary divisors are
{p1(x)e1,1,...,p1(x)e1,k1;p2(x)e2,1,...,p2(x)e2,k2,...,pr(x)er,1, . . . , pr(x)er,kr},
where pi(x), 1 ≤i≤rare distinct prime polynomials, and ei,j are positive
integers. Prove that the minimal polynomial of Ais
m(x) = p1(x)e1·p2(x)e2···pr(x)er,(1)
where ei= max{ei,1,...,ei,ki}.Hint: Let f(x)be the polynomial on the right
hand side of (1). Prove that f(A) = 0. Prove also that pi(x)eidivides the
minimal polynomial m(x).
(b) Conclude, that if the minimal polynomial m(x) of Ais equal to the charac-
teristic polynomial h(x), then the elementary divisors of Aare determined by
h(x). (See the Unique Factorization Theorem 20.18 page 171 in the text).
(c) Let Aand Bbe two 2 ×2 matrices with entries in a field F. Show that A
and Bare similar, if and only if they have the same minimal polynomial.
(d) Let Aand Bbe two 3 ×3 matrices with entries in a field F. Show that A
and Bare similar, if and only if they have the same characteristic polynomial
h(x) and the same minimal polynomial m(x).
(e) Give an example of two 4 ×4 matrices Aand B, which are not similar, but
which have the same characteristic polynomial h(x) and the same minimal
polynomial m(x).
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