Eigenvalues:
Eigenvalues of a matrix A are all the values of π for the following equation:
d = det (πI-A) = 0
For a 2x2 diagonal matrix:
Now,
d = det (πI-A)
π1-a1=0 and π2-a2=0
a1=π1 and a2=π2
Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.
For a 3x3 diagonal matrix:
Now,
d = det (πI-A)
π1-a1=0 ; π2-a2=0 ; π3-a3=0
a1=π1 ; a2=π2 ; a3=π3
Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.