Invers Matriks Positif dan Non-Positif Pertemuan 12 Matakuliah : Matrix Algebra for Statistics Tahun : 2009 Invers Matriks Positif dan Non-Positif Pertemuan 12

M-matrix is a Z-matrix with eigenvalues whose real parts are positive Introduction A nonnegative matrix is a matrix in which all the elements are equal to or greater than zero A positive matrix is a matrix in which all the elements are greater than zero M-matrix is a Z-matrix with eigenvalues whose real parts are positive Bina Nusantara University

A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrix via non-negative matrix factorization A matrix that is both non-negative and positive semidefinite is called a doubly non-negative matrix Bina Nusantara University

Z-matrices The Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, a Z-matrix Z satisfies Bina Nusantara University

Inverse An inverse of a non-singular so-called M-matrix is a non-negative matrix If the non-singular M-matrix is also symmetric then it is called a Stieltjes matrix The inverse of a non-negative matrix is usually not non-negative. An exception is the non-negative monomial matrices Bina Nusantara University

Stieltjes matrix and monomial matrix A Stieltjes matrix is a real symmetric positive definite matrix with nonpositve off-diagonal entries. A Stieltjes matrix is necessarily an M-marix monomial matrix is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Bina Nusantara University

The Banachiewicz Identity and Inverse Positive Matrices An (n−1)×(n−1) real matrix E is given with n ≥2. Assume that E is inverse-positive, i.e., E−1 > 0. Now to this E , one row and one columnare attached to the bottom and to the right end, respectively. Let Bina Nusantara University

where F is in R(n−1)×1, G in R1×(n−1), with ann and being scalars where F is in R(n−1)×1, G in R1×(n−1), with ann and being scalars. Then the Banachiewicz identity gives Where Bina Nusantara University

Example and ann=1 E is not a Z-matrix and Bina Nusantara University

E is inverse-positive Bina Nusantara University