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

3. 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
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4. 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
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5. 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
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6. 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
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7. 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.
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8. 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
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9. 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
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10. Example
and ann=1
E is not a Z-matrix and
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11. E is inverse-positive
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