Invariant Subspaces

Operators and Invariant Subspaces

Definition 76: Operator

A linear map $𝑇 : 𝑉 \to 𝑉$ on a vector space $𝑉$ is called an operator on $𝑉$ . The set of all operators on $𝑉$ is denoted $ℒ (𝑉)$ .

Definition 77: Invariant Subspace

Let $𝑇 \in ℒ (𝑉)$ be an operator on a vector space $𝑉$ . A subspace $𝑊 \subseteq 𝑉$ is invariant under $𝑇$ iff:

𝑇 (𝑊) \subseteq 𝑊 .

That is, for all $𝐰 \in 𝑊$ , we have $𝑇 (𝐰) \in 𝑊$ .

Eigenvalues and Eigenvectors

Definition 78: Eigenvalues and Eigenvectors of Linear Maps

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on a vector space $𝑉$ over a field $𝐹$ . A scalar $𝜆 \in 𝐹$ is called an eigenvalue of $𝑇$ if there exists vector $𝐯 \in 𝑉$ with $𝐯 \neq 𝟎$ such that:

𝑇 (𝐯) = 𝜆 𝐯,

in which case, the vector $𝐯$ is called an eigenvector of $𝑇$ corresponding to the eigenvalue $𝜆$ .

Example.

Let $𝑇 : ℝ^{2} \to ℝ^{2}$ be the linear map defined by $𝑇 (𝑥, 𝑦) = (2 𝑥 + 𝑦, 𝑥 + 2 𝑦)$ . Then $𝜆 = 3$ is an eigenvalue of $𝑇$ because the vector $𝐯 = (1, 1)$ satisfies:

𝑇 (𝐯) = 𝑇 (1, 1) = (2 \cdot 1 + 1, 1 + 2 \cdot 1) = (3, 3) = 3 (1, 1) = 𝜆 𝐯 .

Definition 79: Eigenvalues and Eigenvectors of Matrices

Let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ be a square matrix. A scalar $𝜆 \in 𝐹$ is called an eigenvalue of $𝐴$ if there exists vector $𝐯 \in 𝐹^{𝑛}$ with $𝐯 \neq 𝟎$ such that:

𝐴 𝐯 = 𝜆 𝐯,

in which case, the vector $𝐯$ is called an eigenvector of $𝐴$ corresponding to the eigenvalue $𝜆$ .

Example.

Let $𝐴 = (\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}) \in ℝ^{2 \times 2}$ . Then $𝜆 = 3$ is an eigenvalue of $𝐴$ because the vector $𝐯 = (\begin{matrix} 1 \\ 1 \end{matrix})$ satisfies:

𝐴 𝐯 = (\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}) (\begin{matrix} 1 \\ 1 \end{matrix}) = (\begin{matrix} 3 \\ 3 \end{matrix}) = 3 (\begin{matrix} 1 \\ 1 \end{matrix}) = 𝜆 𝐯 .

Theorem 67: Alternative Characterization of Eigenvalues

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on a vector space $𝑉$ over a field $𝐹$ . A scalar $𝜆 \in 𝐹$ is an eigenvalue of $𝑇$ iff the linear map $𝑇 - 𝜆 {id}_{𝑉}$ is not invertible.

Trace

Definition 80: Trace of a Square Matrix

Let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ be a square matrix. The trace of $𝐴$ , denoted $tr (𝐴)$ , is the sum of the entries on the main diagonal of $𝐴$ :

tr (\begin{matrix} 𝑎_{11} & 𝑎_{22} & \dots & 𝑎_{𝑛 𝑛} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑛 1} & 𝑎_{𝑛 2} & \dots & 𝑎_{𝑛 𝑛} \end{matrix}) = 𝑎_{11} + 𝑎_{22} + \dots + 𝑎_{𝑛 𝑛} = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖 𝑖} .

Theorem 68: Cyclic Property of the Trace

Let $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ and $𝐵 \in 𝐹^{𝑛 \times 𝑚}$ . Then:

tr (𝐴 𝐵) = tr (𝐵 𝐴) .

Definition 81: Trace of a Linear Map

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on a finite-dimensional vector space $𝑉$ over a field $𝐹$ , and suppose $ℬ$ and $𝒞$ are any bases for $𝑉$ . The trace of $𝑇$ , denoted $tr (𝑇)$ , is the trace of the standard matrix ${[𝑇]}_{𝒞 ℬ}$ :

tr (𝑇) = tr ({[𝑇]}_{𝒞 ℬ}) .

Proposition 18: Trace Independence

The trace of a linear map $𝑇 : 𝑉 \to 𝑉$ is independent of the choice of bases for $𝑉$ .

Proof: Proposition 18.

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on a finite-dimensional vector space $𝑉$ over a field $𝐹$ , and let $ℬ$ and $𝒞$ be any bases for $𝑉$ . Let $𝑃 = {[{id}_{𝑉}]}_{𝒞 ℬ}$ be the change-of-basis matrix from the basis $ℬ$ to the basis $𝒞$ . Then:

{[𝑇]}_{𝒞 ℬ} = 𝑃^{- 1} {[𝑇]}_{ℬ ℬ} 𝑃 .

By the cyclic property of the trace, we have:

tr ({[𝑇]}_{𝒞 ℬ}) = tr (𝑃^{- 1} {[𝑇]}_{ℬ ℬ} 𝑃) = tr ({[𝑇]}_{ℬ ℬ} 𝑃 𝑃^{- 1}) = tr ({[𝑇]}_{ℬ ℬ}) .

Therefore, $tr (𝑇)$ is independent of the choice of bases for $𝑉$ .

Theorem 69: Trace of a Linear Map is the Sum of its Eigenvalues

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on a finite-dimensional vector space $𝑉$ over a field $𝐹$ , and let $𝜆_{1}, 𝜆_{2}, \dots, 𝜆_{𝑛}$ be the eigenvalues of $𝑇$ , counted with multiplicity. Then:

tr (𝑇) = \sum_{𝑖 = 1}^{𝑛} 𝜆_{𝑖} .

Exercises

Exercise 73.

Suppose

𝐴 \in 𝐹^{2 \times 2}

. Show that if

tr (𝐴) = 0

, then

𝐴^{2} = 𝑘 𝐼_{2}

for some

𝑘 \in 𝐹

Exercise 74.

Let ${tr}_{𝑛}$ denote the trace function constrained to $𝐹^{𝑛 \times 𝑛}$ .

Show that ${tr}_{𝑛} : 𝐹^{𝑛 \times 𝑛} \to 𝐹$ is a linear map.
Find $\ker ({tr}_{𝑛})$ and $nullity ({tr}_{𝑛})$ .
Let $ℬ_{𝑛}$ be a basis for $𝐹^{𝑛 \times 𝑛}$ defined by $ℬ_{𝑛} = {𝐸_{𝑖 𝑗} \in 𝐹^{𝑛 \times 𝑛} | 1 \leq 𝑖, 𝑗 \leq 𝑛}$ , where $𝐸_{𝑖 𝑗}$ is the matrix with $1$ in the $(𝑖, 𝑗)$ ^th entry and $0$ elsewhere. For example, when $𝑛 = 2$ :
$ℬ_{2} = {𝐸_{11}, 𝐸_{12}, 𝐸_{21}, 𝐸_{22}} = {(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}), (\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}), (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix})} .$
Find ${[{tr}_{𝑛}]}_{ℰ_{1} ℬ_{𝑛}}$ , the standard matrix of ${tr}_{𝑛}$ with respect to the basis $ℬ_{𝑛}$ and the standard basis $ℰ_{1} = {1}$ for $𝐹$ .