ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 3, pp. 258–263.
⃝ Pleiades Publishing, Inc., 2009.
Original Russian Text
⃝ A.V. Seliverstov, V.A. Lyubetsky, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 3, pp. 73–78.
On Symmetric Matrices with
Indeterminate Leading Diagonals
A. V. Seliverstov and V. A. Lyubetsky
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received December 4, 2008; in ﬁnal form, April 10, 2009
Abstract—We consider properties of the matrixof a real quadratic form that takes a constant
value on a suﬃciently large set of vertices of a multidimensional cube centered at the origin
given that the corresponding quadric does not separate vertices of the cube. In particular, we
show that the number of connected components of the graph of the matrixof such a quadratic
form does not change when one edge of the graph is deleted.
We consider a quadratic form ()=
,where runs over the real space of dimension ,
denotes transposition, and is a symmetric real × matrixwith entries
. We assume
throughout what follows that ≥ 3. A quadric given by a pair = ⟨, ⟩ is the aﬃne variety
deﬁned by the equation ()=, being a real number. In fact, we consider the intersection
of a quadric with the set of the -cube whose vertices have coordinates ±1. We denote the
intersection of a quadric and the set by
.Thevariable denotes a diagonal matrix
with arbitrary real numbers on the main diagonal, unless otherwise speciﬁed. We denote by ()
the undirected loop-free graph with no multiple edges and with vertices where the th vertex
is connected by an edge with the th vertexwhenever the entry
of is nonzero. Clearly, the
graph () is independent of diagonal entries of . By rk we denote the rank of a matrix; ∣∣ is
the cardinality of a set .
Properties of () were studied earlier, e.g., what is the smallest rank of a matrixthat deﬁnes
a given graph and how this matrixcan be found . The present paper uses an important theorem
by Fiedler : if (∀)(rk( + ) ≥ − 1), then there exists a permutation matrix such that
is a tridiagonal irreducible matrix. Moreover, this holds over any ﬁeld except for the ﬁeld of
three elements, for which there are some exceptions . Reducibility of a matrix means that it
is block-diagonal, i.e., can be represented as a direct sum of two matrices = ⊕ .
We are interested in the following problems, partial answers to which are given below. Note
that all our proofs, as well as the proof of Fiedler’s theorem in , are elementary.
1. Find a maximum (or minimum) point of a quadratic form () under the constraint ∈
by means of replacing with a “simpler” matrix with the same maximum point. Namely,
we consider the transition from toamatrix + , the maximum point being unchanged:
(∀ ∈ )(∀)[( + )()=()+tr], where tr is the trace of a matrix. Here “simple” means
that in the general case is of a lower rank than . Note that heuristic or approximate (though
eﬃcient) optimization algorithms for matrices of low ranks are known .
2. Derive properties of the graph () from properties of the set
. Among properties
of a graph, one can mention the following: to have at most one connected component (assumed
Supported in part by the International Science & Technology Center, project no. 3807.