theorem for normal triangular matrices


Theorem 1

([1], pp. 82) A square matrixMathworldPlanetmath is diagonal if and only if it is normal and triangular.

Proof. If A is a diagonal matrixMathworldPlanetmath, then the complex conjugateMathworldPlanetmath A is also a diagonal matrix. Since arbitrary diagonal matrices commute, it follows that AA=AA. Thus any diagonal matrix is a normal triangular matrixMathworldPlanetmath.

Next, suppose A=(aij) is a normal upper triangular matrix. Thus aij=0 for i>j, so for the diagonal elements in AA and AA, we obtain

(AA)ii = k=1i|aki|2,
(AA)ii = k=in|aik|2.

For i=1, we have

|a11|2=|a11|2+|a12|2++|a1n|2.

It follows that the only non-zero entry on the first row of A is a11. Similarly, for i=2, we obtain

|a12|2+|a22|2=|a22|2++|a2n|2.

Since a12=0, it follows that the only non-zero element on the second row is a22. Repeating this 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 is a normal upper triangular matrix. Thus, by the above, A is a diagonal matrix, whence also A is a diagonal matrix.

References

Title theorem for normal triangular matrices
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