where:
  • : matrix in the old basis
  • : matrix in the new basis
  • : change of basis matrix
The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
Mnemonic is as follows: consider we have an initial basis . Now, we define the new basis in terms of the old basis, e.g.:
which can be written in matrix form as:
and so if we set:
we have:
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
The answer is that we simply have to calculate the matrix inverse of :
That is the matrix inverse.
When we have a symmetric matrix, a change of bases keeps symmetry iff it is done by an orthogonal matrix, in which case: