Matrix Diagonalisation
The matrix in equation
(12.9) can be written in a different basis such
that it has only nonzero diagonal elements. This is called matrix
diagonalisation. It is a type of similarity transform. See
appendix sec:similarity
for more on similarity transforms. Matrix
diagonalisation is a similarity transform that takes the unit vectors
and (i.e. the basis vectors in which
was written) and turns them into the two eigenvectors. The matrix
that does this similarity transform can be constructed by taking the
column vector eigenvectors to give , i.e.
This is takes the basis states to the eigenvectors. The
eigenvalue equation can now be written in the form
where is the diagonal matrix given in
equation eq:diagonal
. Therefore to calculate the diagonal matrix we
have
so we need to calculate the inverse of
(see section Inverse of a matrix). In our example this is
We can then verify that
as expected.
In summary the general rule for matrix diagonalisation is to create the
similarity transform matrix by collecting the (column) eigenvectors
of , and calculating . eigenvectors are orthonormal, the
similarity transform is unitary, then and
.