Question

Considering the Pauli matrices listed below, construct the similarity transformation S which diagonalizes the matrix.

A = (0 1
1 0)

Answer 1

\[((0 & 1 \ 1 & 0))\]

Answer 2

A and B are similar if there is some matrix P such that B = P*A*inv(P).

Have you heard of eigenvalues and eigenvectors?
They can be used to diagonalize square matrices with n linearly independent eigenvectors.

first, find the eigenvalues of A. They will be -1 and 1.
Then find the corresponding eigenvectors.
For lambda = -1, we get v1 = (1, -1) (column vector)
For lambda = 1, we get v2 = (1, 1) (column vector).

So now let Q be the matrix with columns v1 and v2.
Let D be a diagonal matrix with first entry -1 and second entry 1,
ie. the corresponding eigenvalues to the eigenvectors of Q.

Then A = Q*D*inv(Q).