What are Eigenvectors?

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In studying something, it often helps to start with its definition.

Definition of an Eigenvector

An eigenvector for a linear operator with matrix AA, is a non-zero vector v⃗\vec{v} such that:

Av⃗=λv⃗A\vec{v} = \lambda \vec{v}

for some λ∈R\lambda \in \mathbb{R}.


What does the definition mean?

This means that the matrix AA is scaling v⃗\vec{v}. This means stretching, shrinking, or sometimes even reflecting v⃗\vec{v} to go in the opposite direction.








Problem 1: Finding a Matrix that Scales Everything

Find the right entries for the matrix AA such that AA scales every input v⃗\vec{v} by λ=2.\lambda = 2. Follow the pictures and descriptions below.

Every vector must be doubled by the matrix A, not just some of them.
Every vector must be doubled by the matrix A, not just some of them.


When you have the right matrix entries, enter them here.

Answer Problem 1 correctly to reveal the next part of this lesson.