Permutation-Like Matrices

Permutation Matrices are a well known class of matrices which encode the elements of the symmetric group on $d$ elements as a square $d\times d$ matrix. Motivated by [4], we define a similar class of matrices which are a generalization of Permutation Matrices. We give explicit formulas for the multiplication of these matrices. Lastly, we discuss the spectral radius, eigenvalues, and periodicity before giving a form of Birkhoff-Von Neumann's Theorem for Left Stochastic Matrices.