Inverses and eigenvalues of diamond alternating sign matrices

An n × n diamond alternating sign matrix (ASM) is a (0, +1, −1)-matrix with ±1 entries alternating and arranged in a diamond-shaped pattern. The explicit inverse (for n even) or generalized inverse (for n odd) of a diamond ASM is derived. The eigenvalues of diamond ASMs are considered and when n is even, the characteristic polynomial, which involves signed binomial coe cients, is determined.


Introduction
An n × n alternating sign matrix (ASM) is a ( , + , − )-matrix with + s alternating with − s (ignoring s) such that each row and column begins and ends with + . Research on combinatorial aspects of these matrices, especially on the number of ASMs for a given n, is contained in Bressoud [2]. Here we are concerned with a special class of ASMs.
De nition 1.1. [4] The diamond ASM Dn is the n × n ASM with the ( , n + )-entry equal to +1 and for which the number of nonzero entries in both its rows and columns is given by the vector ( , , , . . . , , , ).
For example, D and D are given by Note that the ± entries form a diamond-shaped pattern.
A matrix A = [a ij ] ∈ R n×n is called centrosymmetric if a ij = a n−i+ ,n−j+ for all ≤ i, j ≤ n. Note that Dn is a symmetric centrosymmetric matrix for all n ≥ . We denote by Pn the n × n antidiagonal matrix It is well known that A ∈ R n×n is centrosymmetric if and only if PnAPn = A. In [3,Theorem 2] it is proved that for a xed n, Dn has the maximum spectral radius over the set of n × n ASMs. This is the only result on the spectra of ASMs that we have found in the literature.
For n even, Dn := PnDn is also an ASM, and is also referred to as a diamond ASM in [4]. For example, D is given by We focus on inverses and spectral properties of diamond ASMs Dn. In Section 2, we give an explicit formula for the inverse (resp. generalized inverse) for n even (resp. n odd). In Section 3, we show that half of the eigenvalues of Dn are ± (resp. 0) for n even (resp. n odd). In the case of n even, by using a recursion, we nd the characteristic polynomial that speci es the remaining eigenvalues of Dn. Interestingly, the coe cients of this polynomial have magnitudes (but not signs) equal to the binomial coe cients.

Inverses . Inverses of D n and D n , n even
Let n = m. By [5, Lemma 2 (i)], Dn may be written in the partitioned form (1) Note that Am + PmBm = Pm.

Furthermore, by [5, Lemma 3 (i)]
where Q is the orthogonal matrix given by Let Sm = diag( , − , . . . , (− ) m+ ). Then where Em is the m × m matrix Using the above results, we now determine the inverse of D m . Proof. Since Am − PmBm and Pm are both invertible, D m is invertible, and using (2) and (3), Substituting (9) into (8) yields the result. (7) gives

Corollary 2.2. For m ≥ , the diamond ASM D m is invertible, and
Note that in D − m the 0 entries form a diamond-shaped pattern.

. Generalized Inverse of D n , n odd
Let n = m + . By [5, Lemma 2 (ii)], Dn may be written in the form where Am is given in (1) where Om is the zero matrix and R is the orthogonal matrix Observe that D m+ has rank m + , so 0 is an eigenvalue of (algebraic and geometric) multiplicity m. Thus the m + nonzero eigenvalues of D m+ are precisely the eigenvalues of the (m + ) × (m + ) matrix Since D m+ is symmetric, the algebraic and geometric multiplicities of the eigenvalue 0 are equal. Thus the group inverse D # m+ exists and furthermore this equals the Moore-Penrose inverse D † m+ , which we now determine.

Theorem 2.4. For m ≥ , the group inverse of the diamond ASM D m+ exists (and is equal to the Moore-Penrose inverse), and
where Proof. From (10) and [1, Theorem 5, p. 140], Letting we now determine Ym, y and ω. The expression in (13) becomes To nd Ym, y and ω, we use (11) and a well-known formula (see, e.g., [6, page 42]) for the inverse of a partitioned matrix to obtain With e T = [ , , · · · , ], Using (14), this completes the proof.

Eigenvalues
For all n ≥ , Dn and Dn have 1 as an eigenvalue with eigenvector [ , , · · · , ] T since all row sums of ASMs are 1.
Let Sn = diag( , − , . . . , (− ) n+ ) and |Dn| be the ( , ) matrix obtained by replacing each entry of Dn by its absolute value. Then SnDnSn = (− ) p |Dn|, where p = if n = k or k + ; p = if n = k + or k + . This leads to the following observation for the spectral radius ρ(Dn). where Am, Bm are as in (1). Furthermore, with Q as in (3), Thus, the eigenvalues of D m are the union of the eigenvalues of Pm, which are 1 with multiplicity m and -1 with multiplicity m , and of (− ) m+ Em. We now give some observations about the eigenvalues of Em, and since numerical computations indicate that some eigenvalues are irrational, we then determine the characteristic polynomial of Em. The following observation is a consequence of (6).

. . Characteristic Polynomials for Em
Let pm(x) = x m + a m− x m− + · · · + a x + a denote the characteristic polynomial of Em, m ≥ . By (15), By Observation 3.2 and (16), Thus the coe cients a i , i = , . . . , m, of pm(x) satisfy To illustrate these relations, we list pm(x) below for small values of m (calculated by Maple): We note that these coe cients coincide with the binomial coe cients up to signs; this motivates Theorem 3.9, for the proof of which we use the following lemmas. The matrix Jm is the m × m matrix with all entries , and 1m is the m-vector with all entries . The following identities can be easily veri ed.
Lemma 3.6. For r ≥ , Proof. Observe that E m+ = (E m+ P m+ )(P m+ E m+ ). In block form this becomes Theorem 3.9. Let pm(x) = x m + a m− x m− + · · · + a x + a be the characteristic polymomial of the m × m nonsingular matrix Em given in (5). Then for q = , , . . ., Proof. Note that pm(x) = det(xIm − Em). We partition xIm − Em into a × block form, and in doing so we distinguish two cases and use mathematical induction. Suppose rst that m = r with r ≥ . For r = and , the result holds with p (x) and p (x) given by (17). Using Lemma 3.5, Using induction, suppose that the result holds for p r (x) and p r− (x). We proceed to nd p r+ (x). Let Then where the multiplication by M T r+ subtracts the second last row from the last row, and the multiplication by M r+ subtracts the second last column from the last column. Now expansion about the last column of this determinant gives the recursion If C[p, x k ] denotes the coe cient of x k from the polynomial p, then for ≤ k ≤ r + , this gives By the induction hypothesis, for even m ≤ r and q = , , . . ., . Eigenvalues of D n , n odd From the discussion in Section 2.2, D m+ has an eigenvalue of multiplicity m, and the m + nonzero eigenvalues of D m+ are precisely the eigenvalues of the (m + ) × (m + ) matrix F m+ that is given in (11). Note that F m+ has 1 as an eigenvalue.
Let q (x) denote the characteristic polynomial of F for ≥ . Calculations using Maple show, for example, that where Am is given by (1) and a, α are given at the beginning of Section 2. The Schur complement on the rst m rows and columns can be used to nd an expression for the characteristic polynomial of F m+ , but we have been unable to nd a recursion (as we used for Em), and thus do not have a general formula for the characteristic polynomial of Dn for n odd.