theorem for normal triangular matrices
Proof. If is a diagonal matrix, then the complex conjugate is also a diagonal matrix. Since arbitrary diagonal matrices commute, it follows that . Thus any diagonal matrix is a normal triangular matrix.
For , we have
It follows that the only non-zero entry on the first row of is . Similarly, for , we obtain
Since , it follows that the only non-zero element on the second row is . Repeating this for all rows, we see that is a diagonal matrix. Thus any normal upper triangular matrix is a diagonal matrix.
Suppose then that is a normal lower triangular matrix. Then it is not difficult to see that is a normal upper triangular matrix. Thus, by the above, is a diagonal matrix, whence also is a diagonal matrix.
|Title||theorem for normal triangular matrices|
|Date of creation||2013-03-22 13:43:35|
|Last modified on||2013-03-22 13:43:35|
|Last modified by||Mathprof (13753)|