The Kronecker product operations in the real linear matrix analytic setting are developed. More versatile operations are proposed that still possess benefits of the usual Kronecker product. Particular attention is paid to the square matrix case motivated by applications to preconditioning and solving systems of linear equations. A number of generically valid factorizations for the inverse are derived. In this connection, approximating an
-linear operator with the arising Kronecker product structures is quite natural and four associated matrix nearness problems are presented. A respective orthogonal decomposition for real matrices with the arising Kronecker singular value decompositions is introduced. Examples of matrix equations leading to
-linear operators with Kronecker product parts are given. Among many ways of solving these equations, a factorization of QZ-type is derived.