On the S-matrix conjecture

Linear Algebra and its Applications 439 (2013) 3555–3560

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Linear Algebra and its Applications www.elsevier.com/locate/laa

On the S-matrix conjecture Roman Drnovšek Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 6 June 2013 Accepted 5 September 2013 Available online 27 September 2013 Submitted by X. Zhan

Motivated with a problem in spectroscopy, Sloane and Harwit conjectured in 1976 what is the minimal Frobenius norm of the inverse of a matrix having all entries from the interval [0, 1]. In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm. His proof is based on the Kiefer–Wolfowitz equivalence theorem from the approximate theory of optimal design. In this note we give a short and simple proof of his result. © 2013 Elsevier Inc. All rights reserved.

MSC: 15A45 15A60 Keywords: Matrices Frobenius norm Inequalities

A Hadamard matrix is a square matrix with entries in {−1, 1} whose rows and hence columns are mutually orthogonal. In other words, a Hadamard matrix of order n is a {−1, 1}-matrix A satisfying A A T = nI , i.e., √1 A is a unitary matrix. n

An S-matrix of order n is a {0, 1}-matrix formed by taking a Hadamard matrix of order n + 1 in which the entries in the first row and column are 1, changing 1’s to 0’s and −1’s to 1’s, and deleting the first row and column. The Frobenius norm of a real matrix A = [ai , j ]ni, j =1 is defined as

 AF =

 n n 

1/2 a2i , j

.

i =1 j =1

It is associated to the inner product defined by

E-mail address: [email protected]. 0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.09.012

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R. Drnovšek / Linear Algebra and its Applications 439 (2013) 3555–3560

n n      A , B  = tr A B T = ai , j b i , j . i =1 j =1

Let Dn denote the set of all matrices A = [ai , j ]ni, j =1 whose entries are in the interval [0, 1]. In 1976, Sloane and Harwit [6] posed the following conjecture. See also [2, p. 59] or [7, Conjecture 11]. Conjecture. If A ∈ Dn is a nonsingular matrix, then

 −1   A   2n , F n+1 where the equality holds if and only if A is an S-matrix. This conjecture arose from a problem in spectroscopy. A detailed discussion of its applications in spectroscopy can be found in [2]. The conjecture has been proved in recent papers [9,8,3] for some special matrices. Apparently, the authors of these papers were not aware of the fact that for odd dimensions the conjecture has already been proved in [1], while for even dimensions a slightly weaker lower bound for the norm has been derived; see [1, Corollary 3.4]. The proof is based on the celebrated equivalence theorem due to Kiefer and Wolfowitz [5] that connects the problem with the approximate theory of optimal design. For an extensive treatment of this theory we refer to [4]. In this note we give a short and transparent proof of the conjecture when n is odd, while for even n our method gives the same (weaker) lower bound as in [1, Corollary 3.4]. Theorem. Let A ∈ Dn be a nonsingular matrix. If n  3 is an odd integer, then

 −1   A   2n , F n+1

(1)

where the equality holds if and only if A is an S-matrix. If n  4 is an even integer, then

√ 2  −1   A  > 2 n − 2n + 2 . F

(2)

n

If n = 2 then

√  −1   A   2, F

(3)



where the equality holds if and only if A is either the identity matrix or

01 10

.

Proof. Let e = (1, 1, 1, . . . , 1) T ∈ Rn and J = ee T . We divide the proof into three cases. Case 1. n  3 is an odd integer, so that n = 2k − 1 for some k ∈ N. Define the matrices M and N of order n + 1 by



M=

1 eT e −k A −1

Since

MNT =

we have

n+1

∗





and

N=

 ∗ , (n + 1) I

1 eT e −(2I − 1k J ) A T



.

R. Drnovšek / Linear Algebra and its Applications 439 (2013) 3555–3560

3557

   M , N  = tr M N T = (n + 1)2 . By the Cauchy–Schwarz inequality, we then obtain

 2 (n + 1)4 =  M , N    M 2F ·  N 2F 

2       1 2  −1 2  = 1 + 2n + k A 1 + 2n +  2I − J A T   . F k

(4)

F

If we show that



    2I − 1 J A T   n,   k

(5)

F

then (4) gives the inequality

 2 (n + 1)2  1 + 2n + k2  A −1  F , and so

 −1  2 n 2 A   = F 2



2

2n n+1

k

completing the proof of (1). To show (5), we determine the maximum of the function f defined on Dn by

 

f ( A) =   2I −

2



 1 = tr A 2I − J J AT   k k F

1



= 4 tr A A =4

 T

n n  

−

a2i , j

2

AT



2k + 1 T = tr 4 A A T − ( Ae )( Ae ) 2 k

2k + 1

( Ae )T ( Ae )  n n 2k + 1  

k2

−

k2

i =1 j =1

i =1

2 ai , j

,

j =1

where A = [ai , j ]ni, j =1 ∈ Dn . Since f is a continuous function on a compact set, it attains its maximum

at some matrix B = [b i , j ]ni, j =1 ∈ Dn . Assume that 0 < b i , j < 1 for some i , j ∈ {1, 2, . . . , n}. Then we have

∂f ( B ) = 0 and ∂ ai , j

∂2 f ( B )  0. ∂ a2i , j

However,

 n  ∂f 2k + 1  ( A ) = 8ai , j − 2 ai ,l , ∂ ai , j k2 l =1

and so

∂2 f 2(2k + 1) 2(4k2 − 2k − 1) ( A) = 8 − = >0 2 2 k k2 ∂ ai , j for all A = [ai , j ]ni, j =1 ∈ Dn . Therefore, we conclude that B is necessarily a {0, 1}-matrix. Let p i be the number of ones in the i-th row of B. Then

f (B) = 4

n  i =1

2k + 1  n

pi −

k2

i =1



2k + 1  n

p 2i = −

k2

i =1

pi −

2k2 2k + 1

2 +

4k2 2k + 1

(2k − 1).

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R. Drnovšek / Linear Algebra and its Applications 439 (2013) 3555–3560 2

Since k − 2k2k+1 = 2kk+1 ∈ (0, 12 ), we have |m − that p i = k for all i. It follows that

f ( B ) = 4nk −

2k + 1 k2

2k2 | 2k+1

> |k −

2k2 | 2k+1

for all m ∈ {1, 2, . . . , n} \ {k}, implying

nk2 = n(4k − 2k − 1) = n2 .

This completes the proof of the inequality (5). Assume that the equality holds in (1). Then there are equalities in (4) and (5), that is, M = N and A is an invertible {0, 1}-matrix with Ae = ke. It follows that k A −1 = (2I − 1k J ) A T = 2 A T − J , and

so e = k A −1 e = 2 A T e − (2k − 1)e implying that A T e = ke. Therefore, we have



NNT = MNT =

1 eT e −k A −1

This means that



NT =

eT J − 2A

1 e





=

n+1 0 0 (n + 1) I



.



eT J − 2A

1 e



is a Hadamard matrix, and so A is an S-matrix. As the equality holds in (1) when A is an S-matrix, the proof is complete for odd dimensions. Case 2. n  4 is an even integer, so that n = 2k for some integer k  2. Define the matrices M and N of order n + 1 by



M=

0 e

Then

√e

MNT =

2k

∗



T

√k k k −1

A −1

∗

k(2k−1) I k −1



and

N=

0 e

√

eT

k√−1 k(2k−1) ( k −1 I k k

− J)A

T

 .

 ,

and so

  2k2 (2k − 1) 2k(2k2 − 1)  M , N  = tr M N T = 2k + = . k−1 k−1 By the Cauchy–Schwarz inequality, we then obtain



2k(2k2 − 1)

2

k−1

 2 =  M , N    M 2F ·  N 2F

 k 3  −1  2  A  = 4k + 4k + g ( A ) , F k−1

where g ( A ) is defined by

√

2   k − 1 k(2k − 1) g( A) =  − J AT  I √  .  k−1 k k

If A =

F

∈ Dn then

(2k − 1)2  T  2 2 (2k − 1)2 g ( A ) = tr A I − J AT = tr A A − ( Ae ) T ( Ae ) k(k − 1) k k(k − 1) k  2k 2 2k 2k 2k 2  (2k − 1)2   2 ai , j − ai , j . = k(k − 1) k [ai , j ]ni, j=1

i =1 j =1

Since

i =1

j =1

(6)

R. Drnovšek / Linear Algebra and its Applications 439 (2013) 3555–3560

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∂2 g 2(2k − 1)2 4 2(4k2 − 6k + 3) ( A) = − = > 0, 2 k(k − 1) k k(k − 1) ∂ ai , j we conclude (similarly as in Case 1) that the maximum of the function g on Dn is attained at some matrix B = [b i , j ]ni, j =1 ∈ Dn with b i , j ∈ {0, 1} for all i and j. Let p i be the number of ones in the i-th row of B. Then

2 2k 2k 2k (2k − 1)2  (2k − 1)2 (2k − 1)4 2 2 2 g(B) = pi − pi = − pi − + . k(k − 1) k k 4(k − 1) 4(k − 1)2 i =1

Since k −

(2k−1)2 4(k−1)

=

− 4(k1−1)

∈

i =1

we obtain that p i = k for all i. It follows that

(2k − 1)2 2k(2k2 − 2k + 1) . 2k − 4k2 = k−1 k−1

g(B) =

Now, since 4k + g ( B ) =

4k +

i =1

(− 12 , 0),

2k(2k2 −1) , k−1

the inequality (6) gives

2 k 3  −1  2  A   2k(2k − 1) , F k−1 k−1

and so

 −1 2 2(2k2 − 2k + 1) 4(n2 − 2n + 2) A   = . F 2 2 k

(7)

n

To complete the proof of the inequality (2), we must exclude the possibility of the equality in (7). So, assume that for some matrix A ∈ Dn the equality holds in (7). Then A is a {0, 1}-matrix and M = N. Therefore, we have

k3 k−1

A



−1

k(2k − 1)

=

or

k−1



k2 2k − 1

I=

I−

I − J AT

k−1 k(2k − 1)

J A T A,

implying that

AT A =

k2



2k − 1

I−

k−1 k(2k − 1)

−1 J

=

k2 2k − 1

I+ T

k−1 k

J .

It follows that the off-diagonal entries of the matrix A A are equal to the number an integer. This is a contradiction with the fact that A is a {0, 1}-matrix. Case 3. n = 2. If



A=

a b c d



is an invertible matrix in D2 , then

A −1 =

1 ad − bc



d −c

−b a



,

and so

 −1  2 a 2 + b 2 + c 2 + d 2 A  = . F (ad − bc )2

k(k−1) 2k−1

that is not

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R. Drnovšek / Linear Algebra and its Applications 439 (2013) 3555–3560

Now, we have

2   (ad − bc )2  A −1  F − 2 = (a − d)2 + (b − c )2 + 2ad(1 − ad) + 2bc (1 − bc ) + 4abcd  0. We conclude that

 −1  2 A   2 F and the equality holds if and only if a = d, b = c, ad ∈ {0, 1}, bc ∈ {0, 1} and abcd = 0. This implies the desired conclusions. 2 Acknowledgement The author was supported in part by the Slovenian Research Agency. References [1] C.-S. Cheng, An application of the Kiefer–Wolfowitz equivalence theorem to a problem in Hadamard transform optics, Ann. Statist. 15 (4) (1987) 1593–1603. [2] M. Harwit, N.J.A. Sloane, Hadamard Transform Optics, Academic Press, New York, 1979. [3] X. Hu, Some inequalities for unitarily invariant norms, J. Math. Inequal. 6 (4) (2012) 615–623. [4] J. Kiefer, General equivalence theory for optimum designs (approximate theory), Ann. Statist. 2 (1974) 849–879. [5] J. Kiefer, J. Wolfowitz, The equivalence of two extremum problems, Canad. J. Math. 12 (1960) 363–366. [6] N.J.A. Sloane, M. Harwit, Masks for Hadamard transform optics, and weighing designs, Appl. Optics 15 (1976) 107–114. [7] X. Zhan, Open problems in matrix theory, in: L. Ji, K. Liu, L. Yang, S.-T. Yau (Eds.), Proceedings of the 4th International Congress of Chinese Mathematicians, vol. I, Higher Education Press, Beijing, 2008, pp. 367–382. [8] L. Zou, On a conjecture concerning the Frobenius norm of matrices, Linear Multilinear Algebra 60 (1) (2012) 27–31. [9] L. Zou, Y. Jiang, X. Hu, A note on a conjecture on the Frobenius norm of matrices, J. Shandong Univ. Nat. Sci. 45 (2010) 48–50.