Functional Properties of the Bergman Kernel in the Space $${{\boldsymbol{\mathbb{C}}}^{\boldsymbol{n}}}\boldsymbol{[m\times m]}$$

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 $${{\mathbb{B}}_{m,n}}$$ .