# Introduction to Eigenvectors

The product of a square matrix **A** and a column vector *v* is a new column vector. The new vector will normally have a different direction from the original, with the matrix representing a *linear transformation*. However, certain vectors will keep their original direction. We say that such a vector is an *eigenvector* of the matrix **A**.

In this article, we will look at eigenvectors, eigenvalues, and the characteristic equation of a matrix. We will also see how to calculate the eigenvectors and values of 2- and 3-dimensional square matrices.

# 2D example

Consider this matrix, **T**:

If we multiply this matrix by the vector (2, 0) we get a new vector (2, 4):

This is illustrated below. The left-hand plot shows the original vector (2, 0) in cyan. It shows several other vectors in different colours. The right-hand graph shows the same set of vectors transformed by the matrix **T** above:

Generally, each transformed vector on the right has a different size and direction compared to its untransformed counterpart on the left.