*Corresponding author’s e-mail: babs3in1@gmail.com ASM Sc. J., 12, Special Issue 5, 2019 for ICoAIMS2019, 72-79 Potential Applications of Hourglass Matrix and its Quadrant Interlocking Factorization Olayiwola Babarinsa 1∗ , Arif Mandangan 2 and Hailiza Kamarulhaili 3 1 Department of Mathematical Sciences, Federal University Lokoja, 1154 Kogi State, Nigeria 2 Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu, Malaysia 1,2,3 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Pulau Pinang, Malaysia Hourglass matrix is recently shown to be a subset of Z-matrix which can be obtained from Quadrant Interlocking Factorization (QIF) of nonsingular matrix. Unlike Z-matrix, the factorization of hourglass matrix may not exist for every nonsingular matrix. However, the potential applications of hourglass matrix and its QIF, such as in statistics (Markov chains), cryptography (GGH encryption scheme) and in graph theory (mixed graph), surpasses the counterpart Z-matrix and its WZ factorization. Lastly, hourglass matrix can be partitioned into triangular block matrices having Schur complement. Keywords: hourglass matrix; z-matrix; quadrant interlocking factorization; markov chains; GGH encryption; mixed graph I. INTRODUCTION The appellation word "hourglass matrix" is coined by Demeure (1989) in describing the matrix derived from factorizing a square matrix, predominantly from real symmetric Toeplitz matrix or Hankel matrix by computing the entries column by column via bowtie-hourglass factorization (WZ factorization or quadrant interlocking factorization (QIF)). However, WZ factorization of nonsingular matrix to yield a butterfly (hourglass) shaped dense square matrix called Z-matrix is first posited by D. Evans and Hatzopoulos (1979). W Z factorization has been modified and applied together with its block factorization being discussed, see for examples (B. Bylina, 2018; D. J. Evans, 2002; Rhofi & Ameur, 2016). Z-matrix exists together with W-matrix during WZ factorization of non- singular matrix , such that (B. Bylina, 2003)  =  (1) Where the entries in  as ℎ , ∗ () =ℎ , ∗ (−1) + , ∗ () ℎ , ∗ (−1) + 1,−+1 ∗ () ℎ −+1, ∗ (−1) (2) and the entries in  are computed from  . () and  .−+1 () as {  . (−1)  . () + −+1. (−1)  .−+1 () = − . (−1)  .−+1 (−1)  . () + −+1.−+1 (−1)  .−+1 () = − .−+1 (−1) (3) For  = 1,2,...,  2 ; ,=  + 1,..., – . The necessary and sufficient condition for matrix  = [ , ] ,=1  to be factorized is that the central submatrices  +2−2  = [ , ] ,=1 +1− are nonsingular, where n is even order of matrix  (the assumption also holds for odd order) and  the centered submatrix of , for  = 1, ...,  2 (Rao, 1997). The factorization is known for the adaptability of its direct method to solve  ×  linear systems given as (Heinig & Rost, 2011).  =  (4) where det() ≠ 0,  = [ 1 , 2 ,…,  ]  , ,   ℝ  ,   ℝ ×  = [ 1 , 2 ,…,  ]   = {  } 1 ≤ ,  ≤ . More so, it was further elucidated that hourglass matrix is the same as Z-matrix which can be partitioned into blocks structured Z-system (J. Bylina & Bylina, 2016; Heinig & Rost, 2005). Unfortunately, there are changes in structure of Z-matrix from  factorization or  which depend on the type of matrix (Toeplitz, Hankel, Hermitian, centrosymmetric, diagonally dominant or tridiagonal matrix) being factorized. Nevertheless, -matrix may not