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A new look at totally positive matrices. (English) Zbl 1413.15059
Previously, 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.
##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A99 Basic linear algebra
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##### References:
  S. M. Fallat, C. R. Johnson: Totally Nonnegative Matrices. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2011.  M. Fiedler: Subtotally positive and Monge matrices. Linear Algebra Appl. 413 (2006), 177–188. · Zbl 1090.15016  M. Fiedler: Remarks on Monge matrices. Math. Bohem. 127 (2002), 27–32. · Zbl 1003.15022  M. Fiedler: Equilibrated anti-Monge matrices. Linear Algebra Appl. 335 (2001), 151–156. · Zbl 0983.15023
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