Hadamard product matrices

A Hadamard Product Involving Inverse Positive Matrices with checkerboard pattern
Manuel F. Abad, Maria T. Gassó and Juan R. Torregrosa
Instituto de Matemática Multidisciplinar UniversidadPolitécnica Valencia

ABSTRACT
A nonsigular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices. In this paper westudy the Hadamard product of inverse-positive matrices whose entries have a particular sign pattern and we show that for any pair of triangular inverse-positive matrices M and N, with checkerboardpattern the Hadamard product M ○ N-1 is closed.

INVERSE-POSITIVE MATRIX
A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative Inverse-positivematrix Inverse-positive matrix Preserved by multiplication, left or right positive diagonal multiplication Preserved by positive diagonal similarity Preserved by permutation similarity It is notpreserved by Hadamard product

←

→

M-matrix M-matrix

⎛ −1 2 ⎞ ⎛0 2 ⎞ A=⎜ ⎟ are inverse-positive matrices, but AºB is not inverse-positive ⎜ 3 −1⎟ B = ⎜ ⎟ ⎜ 3 −1⎟ ⎝ ⎠ ⎝ ⎠
CHECKERBOARD PATTERNAND HADAMARD PRODUCT

SOME INTERESTING ECONOMIC EXAMPLES
a ⎛ −1 ⎜1+ ⎜ a +b 2 ⎜ −1 ⎜ −1 A=⎜ 0 M ⎜ M ⎜ 0 0 ⎜ ⎜ 0 0 ⎝ 0 L 0 −1 L 2 L M 0 L 0 L ⎞ 0⎟ ⎟ 0 0⎟ 0 0⎟ ⎟ M M⎟ 2 −1⎟ ⎟ −1 2 ⎟ ⎠

Definition 1.An n x n real matrix A = (aij) is said to have a checkerboard pattern if sign(aij)=(-1)i+j, i,j = 1, 2,…,n. Definition 2. The Hadamard (or entry-wise) product of two n x n matrices A = (aij) and B =(bij) is A○B=(aijbij). Definition 3. Let A=(aij) be an n x n lower (upper) triangular matrix. A satisfies the P-condition if aij≤aikakj, i > k > j (i < k < j).

is inverse-positive matrix, but A ○ A-1is inverse-positive.

⎛ 1 −a 1 ⎜ 1 −a ⎜ 1 ⎜− a 1 1 A=⎜ M M ⎜ M ⎜ 1 −a 1 ⎜ ⎜− a 1 − a ⎝
⎛ −1 1 0 ⎜ ⎜ 1 −2 1 ⎜ 0 1 −2 A=⎜ M M ⎜M ⎜0 0 0 ⎜ ⎜0 0 0 ⎝

L −a L L L L

1 ⎞ ⎟ 1 − a⎟ −a 1 ⎟ ⎟ M M ⎟ 1...