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[parent] theorem for normal triangular matrices (Theorem)
Theorem 1 ([1], pp. 82)   A square matrix is diagonal if and only if it is normal and triangular.

Proof. If $ A$ is a diagonal matrix, then the complex conjugate $ A^\ast$ is also a diagonal matrix. Since arbitrary diagonal matrices commute, it follows that $ A^\ast A = A A^\ast$. Thus any diagonal matrix is a normal triangular matrix.

Next, suppose $ A=(a_{ij})$ is a normal upper triangular matrix. Thus $ a_{ij}=0$ for $ i>j$, so for the diagonal elements in $ A^\ast A$ and $ AA^\ast$, we obtain

$\displaystyle (A^\ast A)_{ii}$ $\displaystyle =$ $\displaystyle \sum_{k=1}^i \vert a_{ki}\vert^2,$  
$\displaystyle (AA^\ast)_{ii}$ $\displaystyle =$ $\displaystyle \sum_{k=i}^n \vert a_{ik}\vert^2.$  

For $ i=1$, we have
$\displaystyle \vert a_{11}\vert^2 = \vert a_{11}\vert^2+\vert a_{12}\vert^2+\cdots + \vert a_{1n}\vert^2.$
It follows that the only non-zero entry on the first row of $ A$ is $ a_{11}$. Similarly, for $ i=2$, we obtain
$\displaystyle \vert a_{12}\vert^2 + \vert a_{22}\vert^2 = \vert a_{22}\vert^2+\cdots + \vert a_{2n}\vert^2.$
Since $ a_{12}=0$, it follows that the only non-zero element on the second row is $ a_{22}$. Repeating this argument for all rows, we see that $ A$ is a diagonal matrix. Thus any normal upper triangular matrix is a diagonal matrix.

Suppose then that $ A$ is a normal lower triangular matrix. Then it is not difficult to see that $ A^\ast$ is a normal upper triangular matrix. Thus, by the above, $ A^\ast$ is a diagonal matrix, whence also $ A$ is a diagonal matrix. $ \Box$

Bibliography

1
V.V. Prasolov, Problems and Theorems in Linear Algebra, American Mathematical Society, 1994.



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Cross-references: lower triangular matrix, rows, upper triangular matrix, triangular matrix, complex conjugate, diagonal matrix, proof, normal, diagonal, square matrix
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This is version 9 of theorem for normal triangular matrices, born on 2003-06-29, modified 2006-10-02.
Object id is 4412, canonical name is TheoremForNormalTriangularMatrices.
Accessed 5374 times total.

Classification:
AMS MSC15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

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