Consider linear operator T on C2 defined by T zw z iw iz

Consider linear operator T on C² defined by T (z,w) = (z + iw, iz + w). Find a basis such that [T ] is a diagonal matrix.

Solution

T: C²C² is a linear operator defined by T(z,w) = (z+i w, iz+w). Then T(z,w) = A(z,w) where A is the 2x2 matrix with columns (1, i)^T and (i ,1)^T . We shall reduce A to its RREF as under:

Then the RREF of A is a matrix with columns (1, 0)^T and (0,1)^T = B(say)

Then = { ((1, i)^T , (i ,1)^T} and   [T] =   B, the RREF of A which is a diagonal matrix.