Matrices - A. Freddie Page
Matrices are a way of representing linear transformations. As they are written, they are just a list of vectors that tell us where to move a set of basis vectors to during a transformation. I.e., the first column is where the basis vector i maps to, and the second column is where the basis vector j maps to.
Grab the vectors with your mouse, move them around then press “Animate!” to see the transformation in action.
We can see what the transformation does to other vectors by showing it's action on the unit circle. See how the unit circle transforms to an ellipse. Note the colours give an indication of how the unit circle has rotated.
Some matrices have real eigenvectors, these are the vectors whose angle does not change during the transformation. Watch as they are animated, they remain on their span. See how their colour on the unit circle and the transformed ellipse remains the same.
If we know the eigenvalues and eigenvectors of a matrix, we can express it in diagonal form, A = UDU-1. Here we specify which angles are eigendirections, and by how much to scale along each.