Unitary and upper triangular?

**Last_Singularity** · November 24th 2009, 10:04 PM

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

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

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

**tonio** · November 25th 2009, 03:19 AM

Originally Posted by Last_Singularity

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

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

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

You're almost there, though your notation perhaps makes things a little messier:

$U=(u{_ij})$ is unitary, so:

(i) $\Longrightarrow\,U^{*}=U^{-1}$

(ii) $\sum\limits_{k=i}^n(u_{ik}\overline{u_{ik}}=)\left |u_{ik}\right|^2=1$

(iii) The inverse of an upper triangular matrix $U=(u_{ij})$ is also upper triangular with diagonal elements $=u_{ii}^{-1}=\frac{1}{u_{ii}}\,,\,i=1,2,...,n$

Now put together (i) and (iii): $\frac{1}{u_{ii}}=\overline{u_{ii}}\Longleftrightar row\,\left|u_{ii}\right|^2=1$ , and taking into account (ii) this gives us that all the elements outside of the main diagonal must be zero. Q.E.D.

Or simpler, using your reasoning and (iii): $U^{-1}$ is upper triangular, but $U^{-1}=U^{*}$ , and $U^{*}$ is LOWER triangular $\Longrightarrow\,U^{*}$ is both upper and lower triangular and then it is diagonal, and thus also $U$ is.
Tonio

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