  The paper is to reveal the direct links between the well known Sylvester
equation in matrix theory and some integrable systems. Using the Sylvester
equation $\boldsymbol{K} \boldsymbol{M}+\boldsymbol{M}
\boldsymbol{K}=\boldsymbol{r}\, \boldsymbol{s}^{T}$ we introduce a scalar
function $S^{(i,j)}=\boldsymbol{s}^{T}\,
\boldsymbol{K}^j(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^i\boldsymbol{r}$
which is defined as same as in discrete case. $S^{(i,j)}$ satisfy some
recurrence relations which can be viewed as discrete equations and play
indispensable roles in deriving continuous integrable equations. By imposing
dispersion relations on $\boldsymbol{r}$ and $\boldsymbol{s}$, we find the
Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian
Korteweg-de Vries equation and sine-Gordon equation can be expressed by some
discrete equations of $S^{(i,j)}$ defined on certain points. Some special
matrices are used to solve the Sylvester equation and prove symmetry property
$S^{(i,j)}=S^{(i,j)}$. The solution $\boldsymbol{M}$ provides $\tau$ function
by $\tau=|\boldsymbol{I}+\boldsymbol{M}|$. We hope our results can not only
unify the Cauchy matrix approach in both continuous and discrete cases, but
also bring more links for integrable systems and variety of areas where the
Sylvester equation appears frequently.
