[Math] Identity Matrix = Orthogonal

Question

Why is the identity matrix the only n × n matrix that
is orthogonal, upper triangular and has positive entries on the
diagonal? Aren't there other examples where this is the case?

Answer 1

Suppose a matrix, $A$, satisfies orthogonal, upper triangular, and has positive entries.

Then $A^TA=I$, that is $A^T$ is the inverse of $A$, but inverse of upper triangular matrix is upper triangular but $A^T$ is lower triangular, hence $A^T$ must be a diagonal matrix. Hence $A$ is a diagonal matrix.

The eigenvalues of a diagonal matrix are the diagonal entries and we know that the absolute value is $1$ due to orthogonality. We are also told that it has positive entries, hence those diagonal entries are $1$. Hence, $A$ must be the identity matrix.