16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.7.4

Elementary Canonical Forms

Mathematics — Linear Algebra · invariant-direct-sum-det-char-min-poly

Problem

Let

T

be a linear operator on

V

. Suppose

V = W_1 \oplus \cdots \oplus W_k

, where each

W_i

is invariant under

T

. Let

T_i

be the induced (restriction) operator on

W_i

. (a) Prove that

\det(T) = \det(T_1) \cdots \det(T_k)

. (b) Prove that the characteristic polynomial for

T

is the product of the characteristic polynomials for

T_1, \ldots, T_k

. (c) Prove that the minimal polynomial for

T

is the least common multiple of the minimal polynomials for

T_1, \ldots, T_k

. (Hint: Prove and then use the corresponding facts about direct sums of matrices.)