Given a real matrix $A \in \mathbb{R}^{m \times n}$ whose entries are all ones, what is the reduced/short/economy singular value decomposition $A = U\Sigma V^T$? I can see that we have a single singular value for $A$, namely $\sqrt{mn}$, but I'm having trouble coming up with general formulas for the orthogonal matrices $U$ and $V$ in terms of $m$ and $n$.
Singular Value Decomposition of a Real Unit Matrix
linear algebramatricesmatrix decompositionsvd
Best Answer
Suppose $A$ has $k$ nonzero singular values (i.e. $\sigma_k>\sigma_{k+1}=0$). Partition $U$ as $\pmatrix{U_1&U_2}$ and $V$ as $\pmatrix{V_1&V_2}$, where each of $U_1$ and $V_1$ has $k$ columns. The so-called reduced/economic SVD is given by $U_1\operatorname{diag}(\sigma_1,\ldots,\sigma_k)V_1^T$. Note that the result of this matrix product is precisely equal to $A=USV^T$. The terms "reduced" or "economic" refer not to the matrix product, but to the sizes of multiplicands: since $U_1,\operatorname{diag}(\sigma_1,\ldots,\sigma_k)$ and $V_1$ have smaller sizes than $U,\Sigma$ and $V$, it is more economical to store them than to store a full SVD.