Matrix Diagonalisation#

The matrix A 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 x^ and y^ (i.e. the basis vectors in which A 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 S=(u+ u), i.e.

S=(333+15315).

This is takes the basis states to the eigenvectors. The eigenvalue equation can now be written in the form

AS=SA

where A is the diagonal matrix given in equation eq:diagonal. Therefore to calculate the diagonal matrix we have

A=S1AS

so we need to calculate the inverse of S (see section Inverse of a matrix). In our example this is

S1=(3+1561512153+156151215).

We can then verify that

S1AS=(3+1561512153+156151215)(1327)(333+15315)=(4+1500415)

as expected.

In summary the general rule for matrix diagonalisation is to create the similarity transform matrix S by collecting the (column) eigenvectors of A, and calculating S1AS. eigenvectors are orthonormal, the similarity transform is unitary, then S1=S and A=SAS.