  We give a $q$-analog of middle convolution for linear $q$-difference
equations with rational coefficients. In the differential case, middle
convolution is defined by Katz, and he examined properties of middle
convolution in detail. In this paper, we define a $q$-analog of middle
convolution. Moreover, we show that it also can be expressed as a $q$-analog of
Euler transformation. The $q$-middle convolution transforms Fuchsian type
equation to Fuchsian type equation and preserves rigidity index of
$q$-difference equations.
