Solving symmetric matrix word equations via symmetric space machinery.

*(English)*Zbl 1105.15013For a non-empty alphabet \({\mathcal A}=\{A_1, A_2,\dots \}\) the authors consider a semigroup with identity of generalized words of the form \(W:=W(A_1,\dots, A_k)=A_1^{p_1}\cdots A_k^{p_k}\), where \(p_j\) are real numbers. The word \(W\) is symmetric if it is equal to \(A_k^{p_k}\cdots A_1^{p_1}\). A symmetric word equation for \({\mathbf A}={\mathcal A}\cup \{X,B\}\) is an equation of the form \(W(X,A_1,\dots, A_k)=B\), where \(W(X,A_1,\dots, A_k)\) is a symmetric word in \(X,A_1,\dots, A_k\), all exponents of \(X\) are positive, and all exponents of \(A_j\) are non-negative. Symmetric word equations arised naturally in matrix theory as equations over the cone of positive definite matrices. A symmetric word equation \(W(X,A)=B\) is called (uniquely) solvable if there exists a (unique) positive definite solution \(X\) of \(W(X,A)=B\) for any pair of \(n\times n\) positive definite matrices \(A\) and \(B\). It was proved [see C. J. Hillar and C. R. Johnson, Proc. Am. Math. Soc. 132, 945–953 (2004; Zbl 1038.15005)], that every positive definite word equation is solvable.

The authors investigate the uniqueness conjecture and the continuity of solutions as a function on the variables \(A\) and \(B\) over positive definite matrices. The main goal of the paper is threefold.

1. The authors demonstrate how the geometric mean of two matrices and its generalizations to weighted means can be used to give explicit solutions to certain classes of equations.

2. It is shown how the geometry of the positive definite matrices equipped with a symmetric structure and a convex Riemannian metric allows to deduce solution for other classes of symmetric equations via the application of the Banach fixed point theorem.

3. It is shown that equations through degree 5 are uniquely solvable and the solution is continuous in \(A\) and \(B\). It is noted that the degree 5 is the best possible since it was already shown that there are degree 6 equations possessing with multiple solutions.

The authors investigate the uniqueness conjecture and the continuity of solutions as a function on the variables \(A\) and \(B\) over positive definite matrices. The main goal of the paper is threefold.

1. The authors demonstrate how the geometric mean of two matrices and its generalizations to weighted means can be used to give explicit solutions to certain classes of equations.

2. It is shown how the geometry of the positive definite matrices equipped with a symmetric structure and a convex Riemannian metric allows to deduce solution for other classes of symmetric equations via the application of the Banach fixed point theorem.

3. It is shown that equations through degree 5 are uniquely solvable and the solution is continuous in \(A\) and \(B\). It is noted that the degree 5 is the best possible since it was already shown that there are degree 6 equations possessing with multiple solutions.

Reviewer: Alexander Guterman (Moskva)

##### MSC:

15A24 | Matrix equations and identities |

##### Keywords:

symmetric word equation; positive definite matrices; Geometric mean; Non-positive curvature; multiple solutions
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\textit{J. Lawson} and \textit{Y. Lim}, Linear Algebra Appl. 414, No. 2--3, 560--569 (2006; Zbl 1105.15013)

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##### References:

[1] | T. Ando, Topics on Operator Inequalities, Lecture Notes Hokkaido Univ., Sapporo, 1978. |

[2] | Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear algebra appl., 26, 203-241, (1979) · Zbl 0495.15018 |

[3] | Ando, T., On the arithmetic-geometric-harmonic-Mean inequalities for positive definite matrices, Linear algebra appl., 52-53, 31-37, (1983) · Zbl 0516.15011 |

[4] | Ando, T., Majorizations and inequalities in matrix theory, Linear algebra appl., 199, 17-67, (1994) · Zbl 0798.15024 |

[5] | Ando, T.; Li, C.-H.; Mathias, R., Geometric means, Linear algebra appl., 385, 305-334, (2004) · Zbl 1063.47013 |

[6] | S.N. Armstrong, C.R. Hillar, A degree theoretic approach to the solvability of symmetric word equations in positive definite letters, preprint. · Zbl 1134.15009 |

[7] | Bhatia, R., On the exponential metric increasing property, Linear algebra appl., 375, 211-220, (2003) · Zbl 1052.15013 |

[8] | Fujii, M.; Furuta, T.; Nakamoto, R., Norm inequalities in the Corach-Recht theory and operator means, Illinois J. math., 40, 527-534, (1996) · Zbl 0927.47012 |

[9] | Hauser, R.; Lim, Y., Self-scaled barriers for irreducible symmetric cones, SIAM J. optim., 12, 715-723, (2002) · Zbl 1008.90046 |

[10] | Hillar, C.J.; Johnson, C.R., Eigenvalues of words in two positive definite letters, SIAM J. matrix anal. appl., 23, 916-928, (2003) · Zbl 1007.68139 |

[11] | Hillar, C.J.; Johnson, C.R., Symmetric word equations in two positive definite letters, Proc. amer. math. soc., 132, 945-953, (2004) · Zbl 1038.15005 |

[12] | Lang, S., Fundamentals of differential geometry, Graduate text in mathematics, (1999), Springer-Verlag · Zbl 0932.53001 |

[13] | Lawson, J.D.; Lim, Y., The geometric Mean, matrices, metrics, and more, Amer. math. monthly, 108, 797-812, (2001) · Zbl 1040.15016 |

[14] | J.D. Lawson, Y. Lim, Symmetric spaces with convex metric, Forum Math., in press. · Zbl 1169.53333 |

[15] | Lim, Y., Geometric means on symmetric cones, Arch. math., 75, 39-45, (2000) · Zbl 0963.15022 |

[16] | Lim, Y., Applications of geometric means on symmetric cones, Math. ann., 319, 457-468, (2001) · Zbl 1030.17030 |

[17] | Lim, Y., Best approximation in Riemannian geodesic submanifolds of positive definite matrices, Canad. J. math., 56, 776-793, (2004) · Zbl 1067.15020 |

[18] | Maass, H., Siegelâ€™s modular forms and Dirichlet series, Lecture notes in mathematics, vol. 216, (1971), Springer-Verlag Heidelberg · Zbl 0224.10028 |

[19] | Neeb, K.-H., Compressions of infinite-dimensional bounded symmetric domains, Semigroup forum, 61, 71-105, (2001) · Zbl 0980.22005 |

[20] | Nesterov, YU.E.; Todd, M.J., Self-scaled barriers and interior-point methods for convex programming, Math. oper. res., 22, 1-42, (1997) · Zbl 0871.90064 |

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