Geodesic
Properties of a matrix, inverse to a Hessenberg matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part V, Tome 111 (1981), pp. 177-179

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One describes the matrices which are inverses to nonsingular irreducible Hessenberg matrices. 
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     author = {D. K. Faddeev},
     title = {Properties of a matrix, inverse to a {Hessenberg} matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {177--179},
     publisher = {mathdoc},
     volume = {111},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_111_a13/}
}
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D. K. Faddeev. Properties of a matrix, inverse to a Hessenberg matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part V, Tome 111 (1981), pp. 177-179. http://geodesic.mathdoc.fr/item/ZNSL_1981_111_a13/