Section 3.9 Eigenspaces
Checkpoint 3.9.1.
Given \(\mtx{A} = \left[ \begin{array}{rr} 3 \amp 0 \\ 8 \amp -1 \end{array} \right]\) and \(\vect{v} = \op{v_1}{v_2}\text{.}\) Compute \(\mtx{A} \vect{v} - 3\vect{v}\) and \((\mtx{A} - 3 \mtx{I})\vect{v}\text{.}\) What do you notice? What does it mean when \((\mtx{A} - 3 \mtx{I})\vect{v} = \vect{0}\text{?}\) Find at least two vectors \(\vect{v}\) such that \(\mtx{A}\vect{v} = 3\vect{v}\text{.}\)
Definition 3.9.2.
Given a square matrix \(\mtx{A}\text{,}\) when there exists a non-zero vector \(\vect{v}\) such that \(\mtx{A}\vect{v} = \lambda \vect{v}\) for some real number \(\lambda\) that vector is called an eigenvector for \(\mtx{A}\text{,}\) and \(\lambda\) its corresponding eigenvalue.
Example 3.9.3.
Example. The effect of multiplication by a matrix on a representative set of vectors is illustrated below. Shown is a set of vectors and then those same vectors with the result of multiplying each of them by \(\mtx{A} = \left[ \begin{array}{rr} 3 \amp 0 \\ 8 \amp -1 \end{array} \right]\text{.}\) 
Checkpoint 3.9.5.
Given \(\mtx{A} = \left[ \begin{array}{cc} 1 \amp 3 \\ 4 \amp 2 \end{array} \right]\text{,}\) find all values for \(\lambda\) such that the matrix equation
has a solution other that other than \(\vect{v} = \vect{0}\text{.}\) These are the eigenvalues for \(\mtx{A}\text{.}\)
Checkpoint 3.9.6.
Given \(\mtx{A} = \left[ \begin{array}{rrr} 2 \amp 0 \amp 0 \\ 3 \amp -1 \amp 0 \\ 3 \amp 0 \amp -1 \end{array} \right]\) has eigenvalues \(\lambda_\vect{v} = 2\) and \(\lambda_\vect{w} = -1\text{.}\)
Describe the set of vectors \(\vect{v}\) such that \(\mtx{A}\)\(\vect{v}\) = 2\(\vect{v}\)
Describe the set of vectors \(\vect{w}\) such that \(\mtx{A}\)\(\vect{w}\) = -1\(\vect{w}\)
Theorem 3.9.8.
For any \(n \times n\) matrix \(\mtx{A}\text{,}\) the set of eigenvectors corresponding to a given eigenvalue \(\lambda\) becomes a subspace of \(\mathbb{R}^n\) if you include \(\mathbf{0}\text{.}\)