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**Last_Singularity** Question: prove that a matrix that is both unitary and upper triangular must be diagonal.

Call the matrix $\displaystyle A$. By the fact that it is unitary, $\displaystyle AA^* = A^*A = I_n$. In other words, $\displaystyle \sum_{i=1}^n a_{ji} a^*_{ji} = 1$ for all $\displaystyle j=1,...,n$ where $\displaystyle a^*$ is the complex conjugate. Additionally, $\displaystyle \sum_{i=1}^n a_{ji} a^*_{ki} = 0$ for $\displaystyle j \neq k$.

And I know that $\displaystyle A*$ is lower triangular. Any hints, please? Thanks.