Inertia possibilities for completions of partial Hermitian matrices.

*(English)*Zbl 0548.15020The authors define a partial Hermitian matrix A as a one in which some entries are specified and others are left free, and define a completion as a specification of the free entries which gives a Hermitian matrix. In the papers of H. Dym and I. Gohberg [Linear Algebra Appl. 36, 1-24 (1981; Zbl 0461.15002) and of R. Grone, the first author, E. Marques de SĂˇ and H. Wolkowicz [ibid. 58, 109-124 (1984; Zbl 0547.15011)] the problem of when a partial Hermitian matrix can be completed to a positive definite matrix was studied, and a criterion in terms of the (undirected) graph of the specified entries was obtained.

The main result of the present paper is the following generalization to general Hermitian matrices. Let G be a graph with vertex set \(\{\) 1,2,...,\(n\}\) and suppose that G is chordal (that is, its minimal circuits all have lengths at most 3). Let A be a partial Hermitian matrix with a specified entry at the position (i,j) precisely when i and j are adjacent in G. Let i denote the maximum number of nonpositive eigenvalues for any principal submatrix whose rows and columns correspond to a clique in G. Then there is a Hermitian completion of A with exactly i nonpositive eigenvalues. Moreover if each principal submatrix of A corresponding to a clique in G is nonsingular, then a Hermitian completion can be chosen so that it has exactly i negative eigenvalues. Other results deal with the possible inertias of completions, completion to a Toeplitz Hermitian matrix, and the least eigenvalue of suitable completions.

The main result of the present paper is the following generalization to general Hermitian matrices. Let G be a graph with vertex set \(\{\) 1,2,...,\(n\}\) and suppose that G is chordal (that is, its minimal circuits all have lengths at most 3). Let A be a partial Hermitian matrix with a specified entry at the position (i,j) precisely when i and j are adjacent in G. Let i denote the maximum number of nonpositive eigenvalues for any principal submatrix whose rows and columns correspond to a clique in G. Then there is a Hermitian completion of A with exactly i nonpositive eigenvalues. Moreover if each principal submatrix of A corresponding to a clique in G is nonsingular, then a Hermitian completion can be chosen so that it has exactly i negative eigenvalues. Other results deal with the possible inertias of completions, completion to a Toeplitz Hermitian matrix, and the least eigenvalue of suitable completions.

Reviewer: J.D.Dixon

##### MSC:

15B57 | Hermitian, skew-Hermitian, and related matrices |

05C38 | Paths and cycles |

15A18 | Eigenvalues, singular values, and eigenvectors |

##### Keywords:

Chordal graph; partial Hermitian matrix; Hermitian completion; inertias of completions; Toeplitz Hermitian matrix; least eigenvalue
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\textit{C. R. Johnson} and \textit{L. Rodman}, Linear Multilinear Algebra 16, 179--195 (1984; Zbl 0548.15020)

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##### References:

[1] | Dym H., Lin. Alg. Appl 36 pp 1– (1981) · Zbl 0461.15002 · doi:10.1016/0024-3795(81)90215-9 |

[2] | Grone R., Lin. Alg. Appl 58 pp 109– (1984) · Zbl 0547.15011 · doi:10.1016/0024-3795(84)90207-6 |

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