TITLE: Every matrix can be written as a sum of unitary matrices?
QUESTION [20 upvotes]: Any matrix $A \in \mbox{GL}(n, \mathbb{C})$ can be written as a finite linear combination of elements $U_i\in U(n)$:
$$ A = \sum_{i} \lambda_i U_i$$

Is this true? How could I prove it?

REPLY [8 votes]: It is known that every complex square matrix $A$ can be written as a linear combination of at most two unitary matrices. First, by scaling, you may assume that $\|A\|\le1$. Then, by singular value decomposition, you may also assume that
$$
A=\operatorname{diag}(s_1,\ldots,s_n)
$$
where the singular values $s_j$s are real nonnegative and bounded above by $1$. Now, as $s_j=\frac12(z_j+\bar{z}_j)$, where $z_j=s_j+i\sqrt{1-s_j^2}$ has unit modulus, it follows that $A$ is the average of two unitary matrices.
If I remember correctly, there was also an open conjecture that the least possible number $k(n)$ such that every real $n\times n$ matrix can be written as a linear combination of at most $k(n)$ real orthogonal matrices is equal to $4$. It has been shown that $k(n)\le 4$. See proposition 1 in Chi-Kwong Li and Edward Poon, Additive Decomposition of Real Matrices, Linear and Multilinear Algebra, 50(4):321-326, 2002.