S Ü Fen Ed Fak Fen Derg Sayı 22 (2003) 71- 76, KONYA On the Upper Bounds for Permanents Ahmet Ali ÖÇAL1 Abstract: In this paper, considering upper bounds for permanents. λ1 , λ 2 and λ ∞ operator norms, we obtained some Key Words: λ1 , λ 2 and λ ∞ operator norms, permanent Permanentlerin Üst Sınırları Üzerine Özet: Bu çalışmada, λ1 , λ 2 ve λ ∞ operatör normları gözönüne alınarak permanentler için bazı üst sınırlar elde edilmiştir. Anahtar Kelimeler: λ1 , λ 2 ve λ ∞ operatör normları, permanent Introduction and the Statemens of Results Definition 1. [1] The permanent of a real n×n matrix A = (a ij ) is defined by per ( A ) = n ∑∏ a σ∈S n i =1 iσ( i ) , where S n is the symmetric group of order n. Definition 2. ([2]) The λ1 operator norm of an n×n matrix A = (aij ) ∈ Cn×n is defined { } A 1 = max Ax 1 : x ∈ Cn , x 1 = 1 , where x = ( x1 , x 2 ,Κ , x n )T , (T denoting the transpoze) and n x 1 = ∑ xi . i =1 Definition 3. ([2]) The λ2 operator norm of an n×n matrix A = (aij ) ∈ Cn×n is defined A 1 2 { = max Ax 2 : x ∈ Cn , x Department of Mathematics, University of Gazi, 42500 Ankara, TURKEY 2 } =1 , On the Upper Bounds for Permanents where x = ( x1 , x 2 ,Κ , x n )T and 1 x n 2 2 = ∑ x i . i =1 2 Definition 4. ([2]) The λ∞ operator norm of an n×n matrix A = (aij ) ∈ Cn×n is defined A ∞ { = max Ax ∞ : x ∈ Cn , x ∞ } =1 , where x = ( x1 , x 2 ,Κ , x n )T and x ∞ = max x i . 1≤i≤n Lemma 1. Let a1 , a2 ,Κ , an be the columns of A = (aij ) ∈ Cn×n . Then ( n) n per ( A ) ≤ a1 a2 2 2 Κ an 2 , where aj 2 = 1 n ∑a ij i =1 2 2 , 1≤ j ≤n. Proof. We make use of the inequality (see e.g. [1, p.113]) n per ( A ) ≤ ∏ c i i=1 where c 1 , c 2 ,Κ , c n are column sums of A and A = (aij )n×n is a nonnegative matrix. Since per ( A ) ≤ per ( A ) by the triangle inequality, any such bound can be used to produce an upper bound for the permanents of complex matrices. For example from the inequality (1), we obtain n per ( A ) ≤ ∏ q j , j=1 where n q j = ∑ a ij , j = 1, 2 ,Κ , n . i=1 By the Cauchy-Schwarz Inequality, we have 72 (2) Ahmet Ali ÖÇAL n qj = ∑ i =1 aij ≤ 1 n ∑a ij i =1 2 2 1 2 1 = i =1 n ∑ 1 2 2 n ∑a ij i =1 n = aj 2 n. So from inequality (2) we obtain per ( A ) ≤ ( n) n a1 2 a2 2 Κ an 2 and the proof is complete. Theorem 1. Let A 2 { = max Ax 2 : x ∈ C n , x 2 =1 } be λ2 operator norm of A ∈ C n×n . Then n n per ( A ) ≤ n 2 A 2 . Proof. Denote the columns of A by a1 , a2 ,Κ , an and let e1 , e 2 ,Κ , e n be the standart basis of Cn . Then we have aj = A ej , 1≤ j ≤ n . (3) So considering Lemma 1 we have n per ( A ) ≤ n 2 a1 a 2 2 Κ an 2 n 2 ≤ n max a j 2 1≤ j≤n 2 n n 2 = n max Ae j 2 1≤ j≤n ≤ n max Ax 2 x 2 =1 n 2 =n n 2 A n n n 2 and thus the theorem is proved. Lemma 2. Let a1 , a2 ,Κ , an be the columns of A ∈ C n×n . Then 73 On the Upper Bounds for Permanents n per ( A ) ≤ ∏ a j , 1 j=1 where aj n 1 = ∑ a ij , j = 1, 2 ,Κ , n . i=1 Proof. The proof of Lemma is immediately seen from (2). Theorem 2. Let { A 1 = max Ax 1 : x ∈ Cn , x 1 = 1 } be λ1 operator norm of A ∈ C n×n . Then n per ( A ) ≤ A 1 . Proof. Considering Lemma 2 and the equality (3), we have per ( A ) ≤ a1 1 a 2 1 Κ an ≤ Ae1 1 Ae 2 1 Κ Aen ≤ max Ae j 1≤ j≤n 1 ≤ max Ax 1 x =1 1 = A n 1 1 1 n n . Thus the theorem is proved. Lemma 3. Let a1 , a2 ,Κ , an be the columns of A = (a ij ) ∈ C n×n . Then aj 1 ≤ n aj ∞ , where aj 74 n 1 = ∑ a ij , j = 1, 2 ,Κ , n , i=1 Ahmet Ali ÖÇAL and aj = max a ij , 1 ≤ j ≤ n . ∞ 1≤i≤n Proof. For all j, 1 ≤ j ≤ n , we have aj n 1 = ∑ a ij = a1j + a 2 j + Λ + a nj i=1 ≤ n max a ij 1≤i≤n = n aj ∞ . Thus the proof is complete. Theorem 4. Let a1 , a2 ,Κ , an be the columns of A = (a ij ) ∈ C n×n . Then n per ( A ) ≤ n n ∏ a j j=1 ∞ . Proof. Considering Lemma 2 and Lemma 3 the proof is easily seen. Theorem 5. Let A ∞ { = max Ax ∞ : x ∈ Cn , x ∞ =1 } be λ∞ operator norm of A .Then per ( A ) ≤ n n A n ∞ . Proof. From Theorem 4 and equality (3), we have per ( A ) ≤ n n a1 ∞ a2 ∞ Κ an ∞ 75 On the Upper Bounds for Permanents ≤ n max a j 1≤ j≤n n ∞ = n n max Ae j 1≤ j≤n ≤ n max Ax x =1 ∞ n = nn A n ∞ n ∞ ∞ n n , and thus the theorem is proved. REFERENCES 1. Minc, H., Permanents, In Encyclopedia of Mathematics and Its Applications Vol. 6, Addison-Wesley (1978). 2. Taşcı, D., On a Conjecture by Goldberg and Newman, Linear Algebra and its Appl., 215: 275-277 (1995). 76