A new look at totally positive matrices.

*(English)*Zbl 1413.15059Previously, the author [Linear Algebra Appl. 335, No. 1–3, 151–156 (2001; Zbl 0983.15023)]
studied the so-called anti-Monge matrices, namely, matrices \([a_{ik}]\) such that
\[
a_{ij}+a_{kl}\geq a_{il}+a_{kj},
\]
whenever \(i<k\) and \(j<l\). In this note, a relationship is observed between certain anti-Monge matrices and TP\(_2\) matrices. A matrix is called TP\(_2\) if all of the \(1\times 1\) and \(2\times 2\) minors are positive. More generally, a matrix is called totally positive (TP) if all of its minors are positive.

This relationship is described via the Hadamard logarithm (or entry-wise application of the logarithm). That is, the Hadamard logarithm of a TP\(_2\) matrix is a (strict) anti-Monge matrix. Motivated in part by this connection a “multiplication” is defined as follows. Suppose \(X_1\) and \(X_2\) are entry-wise positive square matrices of the same order. Then, the e-product of \(X_1\) and \(X_2\) is defined as: \[ X_1\square X_2=\exp^{\circ}((\log^{\circ}X_1)(\log^{\circ}X_2)), \] where \(\exp^{\circ}\) and \(\log^{\circ}\) are the Hadamard (entry-wise) exponential and logarithm functions, respectively.

With respect to e-multiplication, it is shown that this operation preserves entry-wise positivity and the property of being product-equilibrated (i.e., all row- and column-products are equal to \(1\)). Further along these lines, the author verifies that the operation of e-multiplication preserves a certain sub-class of TP Vandermonde matrices, and also notes that in this special case e-multiplication reduces to the conventional Hadamard product.

This relationship is described via the Hadamard logarithm (or entry-wise application of the logarithm). That is, the Hadamard logarithm of a TP\(_2\) matrix is a (strict) anti-Monge matrix. Motivated in part by this connection a “multiplication” is defined as follows. Suppose \(X_1\) and \(X_2\) are entry-wise positive square matrices of the same order. Then, the e-product of \(X_1\) and \(X_2\) is defined as: \[ X_1\square X_2=\exp^{\circ}((\log^{\circ}X_1)(\log^{\circ}X_2)), \] where \(\exp^{\circ}\) and \(\log^{\circ}\) are the Hadamard (entry-wise) exponential and logarithm functions, respectively.

With respect to e-multiplication, it is shown that this operation preserves entry-wise positivity and the property of being product-equilibrated (i.e., all row- and column-products are equal to \(1\)). Further along these lines, the author verifies that the operation of e-multiplication preserves a certain sub-class of TP Vandermonde matrices, and also notes that in this special case e-multiplication reduces to the conventional Hadamard product.

Reviewer: Shaun M. Fallat (Regina)

##### MSC:

15B48 | Positive matrices and their generalizations; cones of matrices |

15A99 | Basic linear algebra |

##### Keywords:

totally positive matrix; anti-Monge matrix; Vandermonde-like matrix; Hadamard product; equilibrated matrix**OpenURL**

##### References:

[1] | S. M. Fallat, C. R. Johnson: Totally Nonnegative Matrices. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2011. |

[2] | M. Fiedler: Subtotally positive and Monge matrices. Linear Algebra Appl. 413 (2006), 177–188. · Zbl 1090.15016 |

[3] | M. Fiedler: Remarks on Monge matrices. Math. Bohem. 127 (2002), 27–32. · Zbl 1003.15022 |

[4] | M. Fiedler: Equilibrated anti-Monge matrices. Linear Algebra Appl. 335 (2001), 151–156. · Zbl 0983.15023 |

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