PlanetMath: theorem for normal triangular matrices

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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 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 , 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

, 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

is $a_{11}$ . Similarly, for

, 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

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 $A^\ast$ is a normal upper triangular matrix. Thus, by the above, $A^\ast$ is a diagonal matrix, whence also is a diagonal matrix. $\Box$

Bibliography

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

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See Also: normal matrix

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Cross-references: lower triangular matrix, rows, upper triangular matrix, triangular matrix, complex conjugate, diagonal matrix, proof, normal, diagonal, square matrix
There are 3 references to this entry.

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

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