Moore-Penrose inverses of Gram matrices Leaving a Cone Invariant in an Indefinite Inner Product Space

In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.


Introduction
An indefinite inner product in C n is a conjugate symmetric sesquilinear form [x, y] together with the regularity condition that [x, y] = 0 ∀y ∈ C n holds only when x = 0. Associated with any indefinite inner product, there exists a unique invertible hermitian matrix N ∈ C n×n (called a weight) such that [x, y] = x, Ny , where ., . denotes the Euclidean inner product on C n and vice versa. Motivated by the notion of Minkowski space (as studied by physicists), we also make an additional assumption on N, namely, N 2 = I. It should be remarked that this assumption also allows us to compare our results with the Euclidean case, apart from allowing us to present the results with much algebraic ease.
Investigations of linear maps on indefinite inner product spaces employ the usual multiplication of matrices which is induced by the Euclidean inner product of vectors (See for instance [3]). This causes a problem as there are two different values for the dot product of vectors. To overcome this difficulty; Kamaraj, Ramanathan and Sivakumar introduced a new matrix product called indefinite matrix multiplication and investigated some of its properties in [7]. More precisely, the indefinite matrix product of two matrices A and B of sizes m × n and n × l complex matrices, respectively, is defined to be the matrix A • B := ANB. The adjoint of A, denoted by A [ * ] , is defined to be the matrix NA * M, where N and M are weights in the appropriate spaces. Many properties of this product are similar to that of the usual matrix product (refer [7]). Moreover, it not only rectifies the difficulty indicated earlier, but also enables us to recover some interesting results in indefinite inner product spaces in a manner analogous to that of the Euclidean case. Kamaraj, Ramanathan and Sivakumar [7] also shown that in the setting of indefinite inner product spaces, Moore-Penrose inverses of certain matrices do not exist with respect to the usual matrix product where as Moore-Penrose inverses of such matrices exist with respect to the indefinite matrix product. Hence they concluded that indefinite matrix product is more appropriate than the usual matrix product.
The problem of nonnegative invertibility of matrices (or inverses of matrices leaving a cone invariant) was first studied by Collatz [5] when he applied a finite difference method for solving a class of two point boundary value problems. This idea of nonnegative invertibility has undergone a plethora of generalizations over the years. We refer the reader [2] (and the references cited there in) for a detailed survey of these extensions.
In recent years, nonnegative invertibility of Gram matrices has received, a lot of attention. This has been primarily motivated by applications in convex optimization problems. In this connection, there is a well known result that characterizes non negative invertibility of Gram matrices in terms of obtuseness or acuteness of certain polyhedral cones. (See for instance Lemma 1.6 in [4]). Recently, Sivakumar [9] characterized Moore-Penrose inverses of Gram operators leaving a cone invariant over Hilbert spaces. In this paper, we follow the approach of Sivakumar [9] and discuss the Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using indefinite matrix product. As the indefinite matrix product encompasses the Euclidean case as a particular example, it follows that earlier results in the finite dimensional Euclidean spaces, are easy corollaries of our main result.
The paper is organized as follows. In section 2, we introduce basic notations, definitions and results. In section 3, we prove series of lammas and derive the main theorem.

Notations, Definitions and Preliminaries
In this section, we introduce notations, definitions and basic results that will be used in the rest of the paper.
Let ., . denote the usual Euclidean inner product in R n . An indefinite inner product is denoted by [x, y] = x, Ny , where N ∈ R n×n and N = N −1 . Such a matrix N is called weight. A space with an indefinite inner product is called an indefinite inner product space. In the rest of the paper R m , R n denote indefinite inner product spaces with weights M, N respectively. Let A, B be two real matrices of order m × n and n × l respectively, then the indefinite matrix product of those matrices be denoted by A•B and defined as A • B = ANB, where N is a weight matrix as defined earlier. For A ∈ R m×n , the adjoint A [ * ] , of A is defined by A [ * ] = NA * M, where * denotes the transpose of A, M and N are weights of order m and n respectively.
Let K be a subset of R n . Then K is called cone if (i) x, y ∈ K ⇒ x+y ∈ K and (ii)x ∈ K, α ∈ R, α ≥ 0 ⇒ αx ∈ K. The dual of cone K is denoted by K [ * ] and is defined as acute. According to Novikoff, the acuteness of a cone C in R n is defined by the inclusion C ⊆ C * . We can easily verify this condition in indefinite inner product spaces as C ⊆ C [ * ] .
For A ∈ R m×n , A [ * ] • A will be called the Gram matrix of A. For A ∈ R m×n , the following equations are known to have unique solution [7]: Such an X will be denoted by A [ †] . If the weight matrices in indefinite inner product spaces are equal to identity then A [ †] = A † . We refer the reader [1] (and the references cited there in) for a detailed study of A † .
Next, we collect some properties of A [ †] . Some of these have been proved in [7] and rest can be demonstrated easily.
also satisfies the following properties: where R(X) and N (X) denote the range and null spaces of X respectively.
We use the following lemma frequently in this paper.
Lemma 2.2. Let A ∈ R m×n and b ∈ R m . Then, the linear equation A•X = b has a solution iff b ∈ R(A). In this case, the general solution is given by

Main Results
For given A ∈ R m×n , Ramanathan and Sivakumar [8] derived a set of necessary and sufficient conditions for a cone to be invariant under (A [ * ] • A) [ †] . These conditions include pairwise acuteness (or pairwise obtuseness) of certain cones. In this article, we avoid pairwise acuteness of cones and characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in the approach of Sivakumar [9]. These results generalize the existing results of Sivakumar [9] in the finite dimensional setting from Euclidean spaces to indefinite inner product spaces. First we prove series of lemmas that lead up to the main theorem (Theorem 3.16).
As mentioned earlier; R m , R n denote indefinite inner product spaces with weights M, N respectively. Let A ∈ R m×n be such that I • A = A • I that is MA = AN and let K be a closed cone in R n .
Similarly one can easily prove the converse part.
In the next result, we determine the set (A • I • K) [ * ] in the presence of an additional condition. ). This proves the first part.
Next, suppose that only the zero vector. Let y = (1, 2, 0) t ∈ K then The next result is analogous to Theorem 3.4. This will be used later.

Remarks 3.7. Let A be given as in Remark 3.5. Then
. This shows that the condition A [ †] • A • K ⊆ K is essential for the reverse inclusion to hold in Lemma 3.6.

Remarks 3.9. Let A be given as in Remark 3.5. Then
Hence the condition A [ †] • A • K ⊆ K is necessary for the reverse inclusion to hold in Lemma 3.8.
We next obtain a necessary and sufficient condition for a cone to be invariant under (A [ * ] • A) [ †] (See Lemma 3.14).
Conversely, let y = ( We also have stronger one-way implication, given below. The proof follows from necessity part of Lemma 3.11.
Proof. It is enough to show the necessity part. Let x ∈ K [ * ] and y = ( Remarks 3.15. Let A be as given in Remark 3.5 and let K = R 3 + . Then Hence we can conclude that in the absence of the condition A [ †] •A•K ⊆ K, Lemma 3.14 may not be true.
We are now in a position to prove the main result of this article. Then the following conditions are equivalent: (iii) C is obtuse.
Proof.   (iii) For given A ∈ R m×n , Ramanathan and Sivakumar [8] derived a set of necessary and sufficient conditions for a cone to be invariant under (A [ * ] • A) [ †] in terms of pairwise acuteness of cones D and I • D in indefinite inner product space. We would like to remark here that pairwise acuteness of D and I • D is same as acuteness of the cone D in usual inner product space.