This paper provides a general solution for the Kronecker product decomposition (KPD) of vectors, matrices, and hypermatrices. First, an algorithm, namely, monic decomposition algorithm (MDA), is reviewed. It consists of a set of projections from a higher dimension Euclidian space to its factor-dimension subspaces. It is then proved that the KPD of vectors is solvable, if and only if, the project mappings provide the required decomposed vectors. Hence it provides an easily verifiable necessary and sufficient condition for the KPD of vectors. Then an algorithm is proposed to calculate the least square error approximated decomposition. Using it finite times a finite sum (precise) KPD of any vectors can be obtained. Then the swap matrix is used to make the elements of a matrix re-arranging, and then provides a method to convert the KPD of matrices to its corresponding KPD of vectors. It is proved that the KPD of a matrix is solvable, if and only if, the KPD of its corresponding vector is solvable. In this way, the necessary and sufficient condition, and the followed algorithms for approximate and finite sum KPDs for matrices are also obtained. Finally, the permutation matrix is introduced and used to convert the KPD of any hypermatrix to KPD of its corresponding vector. Similarly to matrix case, the necessary and sufficient conditions for solvability and the techniques for vectors and matrices are also applicable for hypermatrices, though some additional algorithms are necessary. Several numerical examples are included to demonstrate this universal KPD solving method.

翻译：本文为向量、矩阵及超矩阵的Kronecker积分解提供了一种通用解法。首先，回顾了一种算法，即首一分解算法，该算法由一组从高维欧几里得空间到其因子维数子空间的投影构成。随后证明，向量Kronecker积分解可解的充要条件是这些投影映射能给出所需的分解向量。由此，为向量的Kronecker积分解提供了一个易于验证的充要条件。接着，提出一种算法来计算最小二乘意义下的近似分解。通过有限次使用该算法，可获得任意向量的有限和（精确）Kronecker积分解。然后，利用交换矩阵对矩阵元素进行重排，进而提供了一种将矩阵的Kronecker积分解转化为其对应向量的Kronecker积分解的方法。证明了矩阵Kronecker积分解可解的充要条件是其对应向量的Kronecker积分解可解。通过这种方式，也获得了矩阵近似分解与有限和分解的充要条件及相应算法。最后，引入置换矩阵，将任意超矩阵的Kronecker积分解转化为其对应向量的分解问题。与矩阵情形类似，尽管需要一些附加算法，但关于可解性的充要条件以及针对向量和矩阵的技术同样适用于超矩阵。文中包含若干数值算例以验证这一通用的Kronecker积分解求解方法。