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:
[tex]Let A =\left[\begin{array}{ccc}a1&0\\0&a2\\\end{array}\right] \\and\\I=\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right][/tex]
[tex]\pi I-A=\left[\begin{array}{ccc}\pi-a1&0\\0&\pi-a2\\\end{array}\right] \\[/tex]
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:
[tex]Let A =\left[\begin{array}{ccc}a1&0&0\\0&a2&0\\0&0&a3\end{array}\right] \\and\\I=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]
[tex]\pi I-A=\left[\begin{array}{ccc}\pi-a1&0&0\\0&\pi-a2\\0&0&\pi-a3\end{array}\right] \\[/tex]
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.
