Abstract
In this paper, we present the Bergman kernel for the matrix ball \({{\mathbb{B}}_{m,n}}\) in the space \({{\mathbb{C}}^{n}}\left[m\times m\right]\) and its functional properties. For this, we use the properties of the automorphism group of the ball \({{\mathbb{B}}_{m,n}}\) . Also, the Bergman kernel is written in the system of complete orthonormal functions in the space \({{\mathbb{C}}^{n}}\left[m\times m\right]\) . Using the reproducing property of the written Bergman kernel, the Bergman integral representation of a function belonging to the Hardy class is written on the ball \({{\mathbb{B}}_{m,n}}\) .
