Eigenvalues and Eigenvectors, Example B

Check if M is hermitian

M=(({{1, 2I}, {-2I, -2}}));{MatrixForm[M],HermitianMatrixQ[M]}

Find the eigenvalues and eigenvectors of M (brute force method)

ML=M-λ IdentityMatrix[2];{Det[ML],lambda=Solve[Det[ML]==0,λ]}

A1=ML/.lambda[[1]];A2=ML/.lambda[[2]];{{MatrixForm[A1],RowReduce[A1]//MatrixForm},{MatrixForm[A2],RowReduce[A2]//MatrixForm}}

Find the eigenvalues and eigenvectors of M (the easy way)

{vals,vecs}=Eigensystem[M];{MatrixForm[vals],MatrixForm[vecs[[1]]],MatrixForm[vecs[[2]]]}

Check if the eigenvectors are orthogonal

vecs[[1]]\[Conjugate].vecs[[2]]

Normalize the eigenvectors

M1=Normalize[vecs[[1]]];M2=Normalize[vecs[[2]]];{MatrixForm[M1],MatrixForm[M2]}

Construct S,and find (SMS^(-1))

Sinv={M1,M2}\[Transpose];S1=Sinv\[ConjugateTranspose];S2=Inverse[Sinv];{MatrixForm[Sinv],MatrixForm[S1],MatrixForm[S2]}

SMSinv=Simplify[S1.M.Sinv];MatrixForm[SMSinv]

Check that det(M)=det((SMS^(-1))), and that Tr(M)=Tr((SMS^(-1)))

{Det[M],Det[SMSinv],Tr[M],Tr[SMSinv]}