|
Consider an n-dimensional
diagonal matrix. We identify n distinct eigenvectors as follows. The
first has 1 for its first component and 0 for the rest of its components.
The second has 1 for its second component and 0 for the rest of its
components. In general, the
 eigenvector has 1 for its
component and 0 for the rest of its components. The
eigenvectors are linearly independent, so they are distinct. An n-dimensional
matrix cannot have more than n distinct eigenvectors, so these are
all the eigenvectors of our matrix. Clearly, the eigenvalue corresponding to
the eigenvector is the
diagonal
element of the matrix. Accordingly, the diagonal elements of the matrix are
its eigenvalues.
[Return]
|
|
