How to prove a unit norm matrix is the average of two unitary matrix
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I assume that the norm is the spectral norm. By the polar decomposition, any unit norm matrix can be written as $UD$ where $U$ is unitary and $D$ is matrix which, in some basis $E$, is diagonal with nonnegative entries not greater than $1$. The diagonal entries can thus be written as $\cos(\theta_1),...,\cos(\theta_n)$ for real $\theta_i$'s. $UD$ is then the average of $U V$ and $U V^*$ where $V$ is the matrix which, in the basis $E$, is diagonal with diagonal entries $e^{i\theta_1},...,e^{i\theta_n}$. 

