The inverse of a block-circulant matrix

The inverse <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A⁻¹</tex> of a block-circulant matrix (BCM) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is given in a closed form, by using the fact that a BCM is a combination of permutation matrices, whose eigenvalues and eigenvectors are found with the help of the complex roots of unity. Special results are also given when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is block symmetric or symmetric.