This work establishes the connection between the finite and infinite algebraic structure of a full row or column rank polynomial matrix and its Moore Penrose inverse. For non-singular matrices, it is known that the inverse has no finite zeros and its finite poles are the finite zeros of the original matrix, while the infinite poles and zeros of the original matrix are the infinite zeros and poles of its inverse respectively. The fact that the Moore Penrose inverse is unique for every matrix along with the fact that for non-singular matrices it coincides with the normal inverse leads to the assumption that a similar connection holds for the structure, both finite and infinite, of the Moore Penrose inverse of a polynomial matrix. We prove that the Moore Penrose inverse of a full row or column rank polynomial matrix has no finite zeros, and that extra poles appear, compared to the non-singular case. Furthermore, we prove that the result regarding the infinite structure of nonsingular matrices is extended to the case of the Moore Penrose inverse of a full rank polynomial matrix. Finally we determine a connection between the minimal indices of the polynomial matrix and its Moore Penrose inverse. The results are also presented through an illustrative example.
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机译：这项工作建立了全行或列秩多项式矩阵的有限和无限代数结构与其Moore Penrose逆之间的联系。对于非奇异矩阵，已知逆没有有限零，并且其有限极点是原始矩阵的有限零，而原始矩阵的无穷极点和零分别是其逆的无限零和极。 Moore Penrose逆对于每个矩阵都是唯一的，对于非奇异矩阵它与法向逆重合，这一事实导致人们假设，Moore Penrose的结构（有限和无限）都具有相似的联系多项式矩阵的逆。我们证明，与非奇异情况相比，全行或列秩多项式矩阵的Moore Penrose逆不具有有限零，并且出现了额外的极点。此外，我们证明了关于非奇异矩阵的无限结构的结果扩展到了全秩多项式矩阵的Moore Penrose逆的情况。最后，我们确定多项式矩阵的最小索引与其摩尔彭罗斯逆矩阵之间的联系。结果还通过一个说明性示例进行了介绍。
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