4.1 An Introduction to eigenvalues and eigenvectors

Kristen Pueschel

4.1 An Introduction to eigenvalues and eigenvectors

Readings

At this time please read Chapter 4 Section 1 in Understanding Linear Algebra by David Austin.

Activity

Please complete the Stretch Factors and Directions Worksheet.

Let A be a matrix. If there is a non-zero vector $\vec{v}$ such that $A \vec{v} = λ \vec{v}$ , for some scalar $λ$ we call $\vec{v}$ an eigenvector and $λ$ its eigenvalue. As $A \vec{0} = \vec{0}$ for all matrices $A$ , we do NOT consider $\vec{0}$ to be an eigenvector.

Eigenvector Facts

All elements of $N u l (A) = {\vec{x} | A \vec{x} = \vec{0}}$ except $\vec{0}$ are eigenvectors of $A$ . Their associated eigenvector is 0. Justify!
If $\vec{v}$ is an eigenvector of matrix A with eigenvalue $λ$ , then $k \vec{v}$ is also an eigenvector with the same eigenvalue for every $k \neq 0$ . Justify!
If $\vec{v}$ is an eigenvector of matrix A with eigenvalue $λ$ , then $\vec{v}$ is an eigenvector for $A^{n},$ with eigenvalue $λ^{n},$ for n=1,2,3,… Check!

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