1. ## orthogonal matrix

two problems that I have problem with:

1. let A be orthogonal matrix that is triangular, prove that A is diagonal.

2.let $
R^n = U_1 + U_2

$
be true, does $
R^n = U_1 ^ \bot + U_2 ^ \bot

$
and if it is why???

2. Originally Posted by omert
two problems that I have problem with:

1. let A be orthogonal matrix that is triangular, prove that A is diagonal.
Say $A$ is upper triangular.

This means $AA^T = I$.
The inverse of a upper triangular matrix is upper triangular.
But $A^T$ is lower triangular.
Thus, $A^T$ is both upper and lower triangular.
Thus, $A^T$ is diagnol which means $A$ is diagnol.