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A Simple introduction to eigenvectors and eigenvalues

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For the matrix

2 -12
1 -5

We will explain what eigenvalues and eigenvectors are as well as verify given eigenvalues and eigenvectors for this matrix.

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Solution Preview

So what is an eigenvalue, or an eigenvector for that matter? These two terms are used by mathematicians to answer one of the most important questions in linear algebra.

When we have a matrix A, is it possible for A to be a scalar multiple of another vector?

This vector we call x, and the scalar lambda.

So is A = lambda(x)

And since any multiple of a vector can be written itself as a multiple of the same vector

Ax = lambda ...

Solution Summary

In this paper, the eigenvalue and eigenvector are presented as a first exposure to students. Definitions are provided, as well as explanations on basic operations using the equation Ax = lambda (x). A matrix and eigenvectors/eigenvalues are given, and the student is asked to verify that these values are correct for the given matrix.