Transforming Matrices

We saw that a transformation matrix P from an old basis S to new basis S' transforms between ()and () as:

A square matrix A transforms similarly, but has in addition the inverse of the transformation matrix at the far right:

The need for two transformation matrices comes from the fact that a matrix provides a transformation of vectors. Given an ``original vector'' , multiplying by matrix A produces an ``image vector'' . When we change coordinates, one transformation matrix is needed to transform , the other to transform :

So the matrix that transforms into is P^-1AP.