Eigenvectors and Upper Triangular Matrices

#LinearAlgebra

Subjects: Linear Algebra
Links: Eigenvalues, Matrix Representation of Linear Transformations, Polynomial Ring of a Single Variable

Def: Let $T \in L (V)$ and $m \in N^{+}$ , then

$T^{m}$ is defined inductively as $T^{0} = I$ and $T^{m + 1} = T \circ T^{m}$
Similarly $T^{- (m + 1)} = T^{- 1} \circ T^{- m}$

Def:************ Let $T \in L (V)$ and $p \in P (F)$ is a polynomial given by:

p (z) = \sum_{k = 0}^{m} a_{k} z^{k}

for $z \in F$ . Then $p (T)$ is the operator defined by:

p (T) = \sum_{k = 0}^{m} a_{k} T^{k}

Def: If $p, q \in P (F)$ , then $p q \in P (F)$ is the polynomial defined by

(p q) (z) = p (z) q (z)

Th: Suppose $p, q \in P (F)$ and $T \in L (V)$ . Then

$(p q) (T) = p (T) q (T)$
$p (T) q (T) = q (T) p (T)$

Th: Every operator on a finite dimensional, nonzero, complex vector space has an eigenvalue.

Th: Suppose $V$ a finite dimensional complex vector space and $T \in L (V)$ . Then $T$ has an upper-triangular matrix with respect to some basis of $V$ .

Th:* Suppose $T \in L (V)$ has an upper-triangular matrix with respect to some basis of $V$ . Then $T$ is invertible iff all the entries on the diagonal of that upper-triangular matrix are nonzero.

Th: Suppose $T \in L (V)$ and $v$ is an eigenvector of $T$ with eigenvalue $λ$ . If $p \in P (F)$ , then $p (T) v = p (λ) v$ .

Th: Suppose $S, T \in L (V)$ and $S$ is invertible. If $p \in P (F)$ , then $p (S T S^{- 1}) = S p (T) S^{- 1}$ .