Answer:
Given:
A triangular matrix is normal if it is diagonal.
For n=2, Let A= [ab0c] such that
A∗A=AA∗= [abbc][ab0c] -[ab0c][a0bc]
= [∣∣a∣∣2+∣∣b∣∣2−∣∣a∣∣2cb−babc−ab∣c∣∣2−(∣∣b∣∣2+∣∣c∣∣2)]
= [b∣∣2cb−babc−ab−∣∣b∣∣2]
So b=0 and A is diagonal ,if the result is true.For n≥2, Let A= [Tv0a] where T is a n x n triangular matrix,v a 1 x n matrix and a is a complex number . Since A∗A=AA∗ we have,
[0000]= [T∗0v∗a][Tv0a]-[Tv0a] [T∗0v∗a]
= [T∗T+vv∗avv∗a∣a∣2]- [TT∗vT∗Tv∗∣v∣2+∣a∣2]
= [T∗T−TT∗+vv∗av−vTv∗a−Tv∗−∣v∣2]
Hence v=0 and T is normal .since T is lower triangular,then T is normal if A is diagonal.
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