theorem for normal triangular matrices
Theorem 1
([1], pp. 82) A square matrix is diagonal if and only if it is normal and triangular.
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.
Next, suppose is a normal upper triangular matrix. Thus for , so for the diagonal elements in and , we obtain
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.
References
- 1 V.V. Prasolov, Problems and Theorems in Linear Algebra, American Mathematical Society, 1994.
Title | theorem for normal triangular matrices |
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Canonical name | TheoremForNormalTriangularMatrices |
Date of creation | 2013-03-22 13:43:35 |
Last modified on | 2013-03-22 13:43:35 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 12 |
Author | Mathprof (13753) |
Entry type | Theorem |
Classification | msc 15A57 |
Classification | msc 15-00 |
Related topic | NormalMatrix |