An algorithm to find values of minors of skew Hadamard and conference matrices C. Kravvaritis† , E. Lappas‡ and M. Mitrouli† † Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece, email:
[email protected] ,
[email protected] ‡ Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece, email:
[email protected] Abstract We give an algorithm to obtain formulae and values for minors of skew Hadamard and conference matrices. One step in our algorithm allows the (n − j) × (n − j) minors of skew Hadamard and conference matrices to be given in terms of the minors of a 2j−1 × 2j−1 matrix. In particular we illustrate our algorithm by finding explicitly all the (n − 3) × (n − 3) minors of such matrices.
1
Introduction
An Hadamard matrix H of order n is an n × n matrix with elements ±1 and HH T = nI. For more details and construction methods of Hadamard matrices we refer the interested reader to the books [1] and [3]. If an Hadamard matrix, H, of order n can be written as H = I + S where S T = −S then H is called skew–Hadamard. A (0, 1, −1) matrix W = W (n, k) of order n satisfying W W T = kIn is called a weighing matrix of order n and weight k or simply a weighing matrix. A W (n, n), n ≡ 0 (mod 4), is a Hadamard matrix of order n. A W = W (n, k) for which W T = −W is called a skew–weighing matrix. A W = W (n, n − 1) satisfying W T = W , n ≡ 2 (mod 4), is called a symmetric conference matrix. Conference matrices cannot exist unless n − 1 is the sum of two squares: thus they cannot exist for orders 22, 34, 58, 70, 78, 94. For more details and construction of weighing matrices the reader can consult the book of Geramita and Seberry [1]. For the conference matrix W (n, n − 1) since W W T = (n − 1)I we have that det(W ) = n (n − 1) 2 . It also holds that the maximum (n − 1) × (n − 1) minors denoted by W (n − 1) are: n n W (n−1) = (n−1) 2 −1 and the maximum (n−2)×(n−2) minors are W (n−2) = 2(n−1) 2 −2 . [2]. In [2] it was also proved that for a matrix A which is skew-Hadamard or conference matrix n n of order n, the (n − 3) × (n − 3) minors are W (n − 3) = 0, 2(n − 1) 2 −3 , or 4(n − 1) 2 −3 for n n n ≡ 0(mod 4) and 2(n − 1) 2 −3 , or 4(n − 1) 2 −3 for n ≡ 2(mod 4). In the present paper we give a useful method for finding the (n − 3) × (n − 3) minors of skew Hadamard and conference matrices. This method points the way to finding other minors such as the (n − j) × (n − j) minors.
1
Notation 1. We use − for −1 in matrices in this paper. Also, when we say the determinants of a matrix we mean the absolute values of the determinants. Notation 2. We write Jb1 ,b2 ,···,bz for the all ones matrix with diagonal blocks of sizes b1 × b1 , b2 × b2 · · · bz × bz . Write aij Jb1 ,b2 ,···,bz for the matrix for which the elements of the block with corners (i + b1 + b2 + · · · + bj−1 , i + b1 + b2 +· · ·+bi−1 ), (i+b1 +b2 +· · ·+bj−1 , b1 +b2 +· · ·+bi ), (b1 +b2 +· · ·+bj , i+b1 +b2 +· · ·+bi−1 ), (b1 + b2 + · · · + bj , b1 + b2 + · · · + bi ) are aij an integer. Write (k − aii )Ib1 ,b2 ,···,bz for the direct sum (k − a11 )Ib1 + (k − a22 )Ib2 + · · · + (k − azz )Ibz .
2
Preliminary Results
We first note a very useful lemma as it allows us to obtain bounds on the column structure of submatrices of a weighing matrix. Lemma 1 (The Distribution Lemma for W (n, n − 1)). Let W be any W (n, n − 1) of n+2 order n > 2. Then, writing ε = (−1) 2 and with a, b, c ∈ {1, −1} for every triple of rows containing 0 a b εa 0 c εb εc 0 the number of columns which are (a) (1, 1, 1)T or (−, −, −)T is 14 (n − 3 − bc − εac − ab) (b) (1, 1, −)T or (−, −, 1)T is 14 (n − 3 − bc + εac + ab) (c) (1, −, 1)T or (−, 1, −)T is 14 (n − 3 + bc − εac + ab) (d) (1, −, −)T or (−, 1, 1)T is 14 (n − 3 + bc + εac − ab) Proof. Let the following rows represent three rows of an weighing matrix W = W (n, n−1) of order n. u1 u2 u3 u4 u5 u6 u7 u8 0 a b 1...1 1...1 1...1 1...1 −...− −...− −...− −...− εa 0 c 1...1 1...1 −...− −...− 1...1 1...1 −...− −...− εb εc 0 1...1 −...− 1...1 −...− 1...1 −...− 1...1 −...− 2
where u1 , u2 , . . . , u8 are the numbers of columns of each type. Then from the order and the inner product of rows we have u1 + u2 + u3 + u4 + u5 + u6 + u7 + u8 = n − 3 u1 + u2 − u3 − u4 − u5 − u6 + u7 + u8 = −bc
(1)
u1 − u2 + u3 − u4 − u5 + u6 − u7 + u8 = −εac u1 − u2 − u3 + u4 + u5 − u6 − u7 + u8 = −ab Solving we have u1 + u8 = u2 + u7 = u3 + u6 = u4 + u5 =
1 (n − 3 − bc − εac − ab) 4 1 (n − 3 − bc + εac + ab) 4 1 (n − 3 + bc − εac + ab) 4 1 (n − 3 + bc + εac − ab) 4 2
Theorem 1 (Determinant Simplification Theorem) Let CC T = (k − aii )Ib1 ,b2 ,···,bz + aij Jb1 ,b2 ,···,bz then det CC T = Πzi=1 (k − aii )bi −1 det D where
D=
k + (b1 − 1)a11 b2 a12 b3 a13 · · · bz a1z b1 a21 k + (b2 − 1)a22 b3 a23 · · · bz a2z .. .. .. .. . . . . b1 az1 b2 az2 b3 az2 · · · k + (bz − 1)azz
(2)
Proof. We note the matrix CC T has k down the diagonal and elsewhere the elements are defined by the block of elements aij . We start with the first row and subtract it from the 2nd to the b1 th row. Then take the first row of the 2nd block (the b1 + 1st row) and subtract it from the b1 + 2th to b1 + b2 th rows. We continue this way with each new block. Now the first column has: a11 − k b1 − 1 times, then a21 followed by b2 − 1 rows of zero, then a31 followed by b3 − 1 rows of zero, and so on until we have az1 followed by bz − 1 rows of zero. We now add columns 2 to b1 to the first column: each of the columns b1 + 2nd to b1 + . . . + b2 th to the b1 + 1st column; and so on until finally add each of the columns b1 + · · · + bz−1 + 2nd to b1 + · · · + bz−1 th to the b1 + · · · + bz−1 + 1st column. The rows which contain zero in the first column will have k − aii on the diagonal and all other elements zero. They can then be used to zero every element in their respective columns. 3
We now expand the determinant, taking into the coefficient those rows and columns which contain k − aii as required. The remaining matrix to be evaluated is D given in the enunciation. 2
3 3.1
The algorithm The matrix Uj
Let xTβ+1 the vectors containing the binary representation of each integer β + 2j−1 for β = 0, . . . , 2j−1 − 1. Replace all zero entries of xTβ+1 by −1 and define the j × 1 vectors uk = x2j−1 −k+1 , k = 1, . . . , 2j−1 We write Uj for all the j × (n − j) matrix in which uk occurs uk times. So u1
z }| {
u2j−1 −1
u2
z }| {
1...1 1...1 1...1 1...1 Uj = . . . . 1...1 1...1 1...1 −...−
... ... ... ... ... ...
u2j−1
u1 u2 1...1 1 1 −...− 1 1 = . . .. .. . . 1 1 −...− 1 − −...−
z }| {
z }| {
1...1 −...− . . 1...1 1...1
. . . u2j−1 −1 u2j−1 ... 1 1 ... − − .. .. . . ... − − ... 1 −
(3)
Example 1 u1 u2 u3 u4 1 1 1 1 U3 = 1 1 − − 1 − 1 − Example 2 u1 u2 u3 u4 u5 u6 u7 u8 1 1 1 1 1 1 1 1 U4 = 1 1 1 1 − − − − 1 1 − − 1 1 − − 1 − 1 − 1 − 1 −
3.2
The matrix D
Any W = W (n, n − 1) matrix can be written: "
W =
M εUjT 4
Uj C
#
,
(4)
where M,C are j ×j and (n−j)×(n−j) matrices respectively, with diagonal entries all 0, such that M = εM T and C = εC T . The elements in the (n − j) × (n − j) matrix CC T obtained by removing the first j rows and columns of the weighing matrix W can be permuted to appear in the form CC T = (n − 1)Iu1 ,u2 ,···,u2j−1 + aik Ju1 ,u2 ,···,u2j−1 where (aik ) = (−ui · uk ), with · the inner product. By the determinant simplification theorem j−1 −j
det CC T = (n − 1)n−2 where D, of order
2j−1
D=
det D
is given by n − 1 − ju1 u2 a12 u3 a13 · · · uz a1z u1 a21 n − 1 − ju2 u3 a23 · · · uz a2z .. .. .. .. . . . . u1 az1 u2 az2 u3 az2 · · · n − 1 − juz
The (n − j) × (n − j) minor of the W (n, n − 1) is the determinant of C for which we have detC = ((n − 1)n−2
j−1 −j
det D)1/2
Remark 1 The algorithm described above works on weighing matrices W (n, n−1) only when all the zeros are on the diagonal, or equivalently the matrix M has j zeros and the C matrix has (n-j) zeros. Consequently it might loose some expected values for some minors.
4
Application of the algorithm on the (n-3)×(n-3) minors of weighing matrices
The matrix (aik ) = (−ui · uk ) is
(aik ) =
4.1
−3 −1 −1 1 −1 −3 1 −1 −1 1 −3 −1 1 −1 −1 −3
Case 1, ε=1
Without loss of generality we can assume that the matrix M is
0 1 1 M = 1 0 1 1 1 0 then by the Distribution Lemma u1 =
n−6 4 ,
u2 =
n−2 4 ,
u3 =
n + 14 −n + 2 −n + 2
−n + 6 1 D= 4 4 −n + 6
n−2 4 ,
u4 =
n−2
n+2
n − 2 −n + 2
n−2
n + 2 −n + 2
n − 6 −n + 2 −n + 2 5
n+2
n−2 4 .
The matrix D is:
and its determinant is detD = 1024n − 1024. Finally detCC T = (n − 1)n−7
4.2
1024n − 1024 256
Case 2, ε=-1
Without loss of generality we can assume that the matrix M is
0 1 1 M = −1 0 1 −1 −1 0 then by the Distribution Lemma u1 =
1 D= 4 4
n−4 4 ,
u2 =
n−4 4 ,
n + 8 −n + 4
−n + 4 −n + 4
n+8
u3 = n4 , u4 = −n
n−4
n−4 4 .
The matrix D is:
n −n + 4
n − 4 n − 4 −n + 4
n − 4 −n + 4
−n
n+8
and its determinant is detD = 0. Finally detCC T = 0
5
Application to Numerical Analysis
Let A = [aij ] ∈ Rn×n . We reduce A to upper triangular form by using Gaussian Elimination (k) (GE) operations. Let A(k) = [aij ] denote the matrix obtained after the first k pivoting operations, so A(n−1) is the final upper triangular matrix. A diagonal entry of that final matrix will be called a pivot. Matrices with the property that no exchanges are actually needed during GE with complete pivoting are called completely pivoted (CP) or feasible. (k) (0) Let g(n, A) = max |aij |/|a11 | denote the growth associated with GE on a CP A and g(n) = i,j,k
sup{ g(n, A)/A ∈ Rn×n }. The problem of determining g(n) for various values of n is called the growth problem. The determination of g(n) remains a mystery. Wilkinson in [5] proved that g(n) ≤ [n 2 31/2 . . . n1/n−1 ]1/2 = f (n) Wilkinson’s initial conjecture seems to be connected with Hadamard matrices. Interesting results in the size of pivots appears when GE is applied to CP skew-Hadamard and conference matrices of order n. In these matrices, the growth is also large, and experimentally, we have been led to believe it equals n − 1 and special structure appears for the first few and last few pivots. These results give rise to new conjectures that can be posed for this category of matrices. The growth conjecture for skew Hadamard and conference matrices Let W be a CP skew-Hadamrad or conference matrix. Reduce W by GE. Then 6
(i) g(n, W ) = n − 1. (ii) The two last pivots are equal to
n−1 2 ,n
− 1.
(iii) Every pivot before the last has magnitude at most n − 1. (iv) The first four pivots are equal to 1, 2, 2, 3 or 4, for n > 14. The magnitude of the pivots appearing after GE operations on a CP matrix W is pj =
W (j) , j = 1, . . . , n, W (0) = 1 W (j − 1)
(5)
This relationship gives the connection between pivots and minors, so it is obvious that the calculation of minors is very important. Results • The first 4 pivots for every W (n, n − 1) 1, 2, 2, 3 or 4 • The 3 last pivots for every W (n, n − 1) n−1 n − 1 or n−1 2 , 2 ,n − 1 • Pivot patterns of W (8, 7): {1, 2, 2, 4, 7/4, 7/2, 7/2, 7} or {1, 2, 2, 3, 7/3, 7/2, 7/2, 7}. • Pivot patterns of W (10, 9): {1, 2, 2, 3, 3, 4, 9/4, 9/2, 9/2, 9} or {1, 2, 2, 4, 3, 3, 9/4, 9/2, 9/2, 9} or {1, 2, 2, 3, 10/4, 18/5, 9/3, 9/2, 9/2, 9}.
References [1] A.V.Geramita, and J.Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979. [2] C. Koukouvinos, M. Mitrouli and J. Seberry, Growth in Gaussian elimination for weighing matrices, W (n, n − 1), Linear Algebra and its Appl., 306 (2000), 189-202. 7
[3] J. Seberry Wallis, Hadamard matrices, Part IV, Combinatorics: Room Squares, SumFree Sets and Hadamard Matrices, Lecture Notes in Mathematics, Vol. 292, eds. W. D. Wallis, Anne Penfold Street and Jennifer Seberry Wallis, Springer-Verlag, BerlinHeidelberg-New York, 1972. [4] F. R. Sharpe, The maximum value of a determinant, Bull. Amer. Math. Soc., 14 (1907), 121-123. [5] J. H. Wilkinson, Error analysis of direct methods of matrix inversion, J. Assoc. Comput. Mach., 8 (1961), 281-330.
8