Linear Algebra Done Right · Problem 5C.10

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

T \in \mathcal{L}(V)

and

v_1, \ldots, v_n

is a basis of

V

. Show that the following are equivalent. (a) The matrix of

T

with respect to

v_1, \ldots, v_n

is lower triangular. (b)

\operatorname{span}(v_k, \ldots, v_n)

is invariant under

T

for each

k = 1, \ldots, n

. (c)

Tv_k \in \operatorname{span}(v_k, \ldots, v_n)

for each

k = 1, \ldots, n

. A square matrix is called lower triangular if all entries above the diagonal are

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