Norm optimal factorizations of scalar and block matrices

For an $m\times n$ complex matrix $X$ of rank $r$ with Schur multiplier $S_X$ we show that there exist an $r\times m$ complex matrix $L$ and an $r\times n$ complex matrix $R$ such that $X=L^{\ast }R$ and $\mathrm{\parallel }{S}_X\mathrm{\parallel }=\mathrm{\parallel }\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(L^{\ast }L){\mathrm{\parallel }}^{\frac{1}{2}}\mathrm{\parallel }\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(R^{\ast }R){\mathrm{\parallel }}^{\frac{1}{2}},$ and the norm condition is optimal. Let the completely bounded norm of the bilinear form $B_X$ induced by $X$ on $({\mathbb{C}}^m,\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_{\mathrm{\infty }})\times ({\mathbb{C}}^n,\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_{\mathrm{\infty }})$ be denoted $\mathrm{\parallel }{B}_X{\mathrm{\parallel }}_{cb},$ then $X$ has a factorization $X=\mathrm{\Delta }(\eta {)}^{\ast }C\mathrm{\Delta }(\xi )$ with $\eta$ in ${\mathbb{C}}^m,$ $\xi$ in ${\mathbb{C}}^n$ such that the outer factors are diagonal operators with $\mathrm{\parallel }\xi {\mathrm{\parallel }}_2=\mathrm{\parallel }\eta {\mathrm{\parallel }}_2=1$ and $C$ has operator norm equal to $\mathrm{\parallel }{B}_X{\mathrm{\parallel }}_{cb},$ and the norm condition is optimal. A generalization to operator valued Schur block multipliers is presented too.