Linear Algebra Done Right · Problem 5C.8

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

V

is finite-dimensional,

T \in \mathcal{L}(V)

, and there exists a nonzero vector

v \in V

such that

T^2v + 2Tv = -2v

. (a) Prove that if

\mathbf{F} = \mathbf{R}

, then there does not exist a basis of

V

with respect to which

T

has an upper-triangular matrix. (b) Prove that if

\mathbf{F} = \mathbf{C}

and

A

is an upper-triangular matrix that equals the matrix of

T

with respect to some basis of

V

, then

-1 + i

-1 - i

appears on the diagonal of

A