Kronecker products, vec operator, and Moore‐Penrose inverse
Jan R. Magnus, Jan R. Magnus · Wiley series in probability and statistics · 2019
This chapter develops some matrix tools that will prove useful to us later. The vec operator transforms a matrix into a vector by stacking its columns one underneath the other. The chapter shows that the Kronecker product and the vec operator are intimately connected. It discusses the Moore-Penrose (MP) inverse, which generalizes the concept of the inverse of a nonsingular matrix to singular square matrices and rectangular matrices. The inverse of a matrix is defined when the matrix is square and nonsingular. For many purposes, it is useful to generalize the concept of invertibility to singular matrices and, indeed, to matrices that are not square. One such generalization that is particularly useful because of its uniqueness is the Moore-Penrose inverse.