16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.2.21

The Rational and Jordan Forms

Mathematics — Linear Algebra · diagonalizable-complex-operator

Problem

Let

A

be an

n \times n

matrix with real entries. Let

T

be the linear operator on

R^n

which is represented by

A

in the standard ordered basis, and let

U

be the linear operator on

\mathbb{C}^n

which is represented by

A

in the standard ordered basis. Use the result of Exercise 20 to prove the following: If the only subspaces invariant under

T

are

R^n

and the zero subspace, then

U

is diagonalizable.