The idea of this applet is to help to understand what Diagonalising a matrix achieves.
It is my first attempt at writing anything in Geogebra, so apologies for the glitches
A is a transformation, which you can edit by double clicking on it.
The applet will need reloading if A is a two way stretch along the x and y axes (i.e. if opposite entries are 0).
The normalised matrix of eigenvectors (P) and its inverse (Pinv) are calculated.
The Diagonalised matrix is also calculated.
Note that if you are working from the Edexcel syllabus, they only use symmetric matrices for A and the inverse of P will be its transpose.
The transformation P is applied to the co-ordinate system (the new grid shows the new Axes X' and Y' and a grid based on these axes at the same intervals as whatever the X,Y grid is showing; the X,Y grid is switched off but can be switched on by right clicking on the screen)
The transformation A can be considered as a stretch along the new X' axis and a stretch along the new Y' axis with scale factors given by the values in the Diagonalised matrix.
If you grab point L and drag it around, you will see its coordinates in the new system displayed as r'.
Pick L up, take it to an object point, and note down r'.
Now move it to the image, and note down the new r'.
You should see how the new x' and y' coordinates are simply the object x' and y' coordinates multiplied by the eigenvalues- showing the stretch clearly.
Enjoy; you can break it by using non-diagonalisable transformations or getting eigenvectors that are parallel to the x and y axes!