# Linear algebra demos

## The basics

Check out my CoCalc linear algebra primer (PDF) to refresh your knowledge on vectors and matrices.

## Eigenvalues and eigenvectors

When a vector is multiplied by a matrix from the left, then in general it will be rotated and scaled (made "longer" or "shorter"). The "eigenvectors" of a matrix (from the German word "eigen", which means "own") have the special property that they are not rotated by the matrix they belong to, only scaled. The scaling factors are called the "eigenvalues".

In this demonstration we apply the matrix

, which has eigenvalues 5 and 2, to a unit vector *x* that sweeps the whole
unit circle. We will see that the transformed result sweeps out an ellipsoid, and
in 2 cases it points into the same direction as the input vector, which are the
eigenvectors of the matrix * A*.

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