Introduction to Eigenvectors

Martin McBride
Technological Singularity
9 min readApr 19, 2024

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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.

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