Geodesic
Matrix Quadratic Equations, Column/row Reduced Factorizations and an Inertia Theorem for Matrix Polynomials
International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 6, pp. 1285-1310.

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It is shown that a certain Bezout operator provides a bijective correspondence between the solutions of the matrix quadratic equation and factorizatons of a certain matrix polynomial G(lambda) (which is a specification of a Popov-type function) into a product of row and column reduced polynomials. Special attention is paid to the symmetric case, i.e. to the Algebraic Riccati Equation. In particular, it is shown that extremal solutions of such equations correspond to spectral factorizations of G(lambda). The proof of these results depends heavily on a new inertia theorem for matrix polynomials which is also one of the main results in this paper.
Keywords: matrix quadratic equations, Bezoutians, inertia, column (row) reduced polynomials, factorization, algebraic Riccati equation, extremal solutions
Mots-clés : macierz, równanie różniczkowe Riccatiego 
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     author = {Karelin, I. and Lerer, L.},
     title = {Matrix {Quadratic} {Equations,} {Column/row} {Reduced} {Factorizations} and an {Inertia} {Theorem} for {Matrix} {Polynomials}},
     journal = {International Journal of Applied Mathematics and Computer Science},
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Karelin, I.; Lerer, L. Matrix Quadratic Equations, Column/row Reduced Factorizations and an Inertia Theorem for Matrix Polynomials. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 6, pp. 1285-1310. http://geodesic.mathdoc.fr/item/IJAMCS_2001_11_6_a4/