published in Linear Algebra and its Applications (LAA), Vol. 363, 237-250, 2003. ON P -MATRICES SIEGFRIED M. RUMP ¤ Abstract. We present some necessary and su±cient conditions for a real matrix being P -matrix. They are based on the sign-real spectral radius and regularity of a certain interval matrix . We show that no minor can be left out when checking for P -property. Furthermore, a not necessarily exponential method for checking P -property is given. 1. Introduction and notation. A real matrix A 2 Mn(IR) is called P -matrix if all its principal minors are positive. The class of P -matrices is denoted by P. The P -problem, namely the problem of checking whether a given matrix is a P -matrix, is important in many applications, see [1]. A straightforward algorithm evaluating the 2n ¡ 1 principal minors requires some n32n operations. This corresponds to the fact that the P -problem is NP -hard [2]. In Theorem 2.2 we will show that none of these minors can be left out. However, there are other strategies. Recently, Tsatsomeros and Li [20] presented an algorithm based on Schur complements reducing computational complexity to 7 ¢ 2n. The algorithm requires always this number of operations if the matrix in question is a P -matrix. Otherwise, the computational cost is frequently much smaller because one nonpositive minor su±ces to prove A2 = P. In this paper we will present characterizations of P -matrices related to the sign-real spectral radius, and based on that some necessary conditions and su±cient conditions. In case A2 = P we also derive strategies to ¯nd a nonpositive minor. Finally, we give an algorithm which is not a priori exponential for A 2 P, but can be so in the worst case. The method is tested for n = 100, where all other known methods require 2100 operations. However, this approach needs further analysis. We use popular notation in matrix theory. Especially, A[¹] denotes the principal submatrix of A with rows and columns out of ¹ ⊆ f1; : : : ; ng. Absolute value and comparison of vectors and matrices is always to be understood componentwise. For example, signature matrices S are characterized by jSj = I. 2. Characterization of P -property. In [17] we introduced and investigated the sign-real spectral S radius ½0 . In the meantime we also introduced the sign-complex spectral radius. Therefore, for better readability, we changed the notation into ½IR(A) for the sign-real and ½C(A) for the sign-complex spectral radius. The sign-real spectral radius is de¯ned by (1) ½IR(A) := maxfj¸j : SAx = ¸x; jSj = I; 0 6= x 2 IRn; ¸ 2 IRg Note that the maximum is taken over the absolute values of real eigenvalues. Among the characterizations given in [17] is the following [Theorem 2.3]. For 0 < r 2 IR, (2) ½IR(A) < r , det(rI + SA) > 0 for all jSj = I (3) , det(rI + DA) > 0 for all jDj · I: This leads to two characterizations of the P -property. Theorem 2.1. For A 2 Mn(IR) and a positive r such that det(rI ¡ A) 6= 0 the following are equivalent: (i) C := (rI ¡ A)¡1(rI + A) 2 P. (ii) ½IR(A) < r. For nonsingular A, parts (i) and (ii) are equivalent to ¡1 ¡1 ¡1 ¡1 (iii) All B 2 Mn(IR) with A ¡ r I · B · A + r I are nonsingular. ¤ Inst. f. Informatik III, Technical University Hamburg-Harburg, Schwarzenbergstr. 95, 21071 Hamburg, Germany 1 Remark. The assertions follow by [17, Theorem 2.13 and Lemma 2.11]. Following, we give di®erent and simpler proofs. This also allows to conclude the subsequent Theorem 2.2. As remarked by one referee, the assertions also follow by (2), (3) and [9, Theorem 3.4], see also [10, 18]. Proof. Let a ¯xed but arbitrary signature matrix S be given and de¯ne ¹ ⊆ f1; : : : ; ng by (4) ¹ := fi : Sii = 1g 1 De¯ne diagonal D by D := 2 (I ¡ S), so that S = I ¡ 2D and Dii = 0 for i 2 ¹; Dii = 1 for i2 = ¹. Then (I ¡ D)C + D comprises of the rows of C out of ¹, and the rows of the identity matrix out of f1; : : : ; ng n ¹. Therefore, (5) det((I ¡ D)C + D) = det C[¹]: On the other hand, C = (rI ¡ A)¡1(rI + A) = (rI + A)(rI ¡ A)¡1 and (I ¡ D)C + D = f(I ¡ D)(rI + A) + D(rI ¡ A)g(rI ¡ A)¡1 = frI + A ¡ 2DAg(rI ¡ A)¡1 = (rI + SA)(rI ¡ A)¡1; and in view of (5), (6) C 2 P , 8jSj = I : det(rI + SA)= det(rI ¡ A) > 0: Now X det(rI + SA) = det(SA)[!] ¢ rn¡j!j; ! where the sum is taken over all ! ⊆ f1; : : : ; ng including ! = ;. Summing the determinants over all S, all terms cancel except for ! = ;, such that X det(rI + SA) = 2n ¢ rn: jSj=I Therefore, not all det(rI + SA); jSj = I, can be negative. This implies with (6), C 2 P , 8jSj = I : det(rI + SA) > 0; and proves (i) , (ii). Concerning (iii), we use characterization (3) and a continuity argument to obtain ½IR(A) ¸ r , 9 jDj · I : det(rI + DA) = 0 , 9 jDej · r¡1I : det(A¡1 + De) = 0: As a result of the previous proof we have a one-to-one correspondence between the minors of C and signature matrices S in (5) and (6): For det(rI ¡ A) > 0, det C[¹] > 0 , det(rI + SA) > 0(7) for ¹ as de¯ned in (4). As a result we obtain a solution to a question posed at our meeting in Oberwolfach. Theorem 2.2. For every n ¸ 2 and every ; 6= ¹ ⊆ f1; : : : ; ng, there exists a matrix C 2 Mn(IR) with det C[¹] < 0; and det C[!] > 0 for all ! ⊆ f1; : : : ; ng;! 6= ¹: Proof. De¯ne B := (1) 2 Mn(IR), the matrix all components of which are 1's. Obviously, ½IR(B) = ½(B) = n. For every jSj = I, SB is of rank 1, so that the characteristic polynomial of SB is 2 n n¡1 ÂSB(x) = det(xI ¡ SB) = x ¡ tr(SB) ¢ x . Therefore, ÂSB(x) is positive for x > max(0; tr(SB)). But tr(SB) · n ¡ 2 for all jSj = I, S 6= I, and tr(B) = n. Hence, for every n ¡ 2 < r < n, (8) det(rI ¡ B) < 0; and det(rI ¡ SB) > 0 for all jSj = I;S 6= I: Let n ¸ 2 and ; 6= ¹ ⊆ f1; : : : ; ng be given. De¯ne jS0j = I by ( 1 for i 2 ¹ S0 = ii ¡1 otherwise; and set A := ¡S0B. For ¯xed r; n ¡ 2 < r < n, de¯ne C := (rI ¡ A)¡1(rI + A) . Then S0 6= ¡I because ¹ 6= ;, and det(rI ¡ A) > 0 by (8). Furthermore, by (8), S 6= S0 ) det(rI + SA) = det(rI ¡ SS0B) > 0; S = S0 ) det(rI + SA) = det(rI ¡ B) < 0: Finally, the equivalence (7) ¯nishes the proof. IR 0 The proof relies on the following fact. Let A 2 Mn(IR) and r := ½ (A). Then there is r < r with det(r0I ¡ SAe ) < 0 for some jSej = I, and det(r0I ¡ SA) > 0 for all jSj = I, S 6= Se. This is explored in the proof for a speci¯c matrix. We mention that, due to numerical experience, this seems by no means a rare case but rather typical for generic A and r0 · r, r0 ¼ r. 3. Necessary and su±cient conditions. In this section we present conditions for testing the P -property for a given matrix C 2 Mn(IR). First we make sure that the spectral radius of C is less than one. Set dlog ®e (9) ® = kCk1 + 1; ¯ = 2 2 ; C = C=¯; The P -property of C is not changed by the scaling; so we may assume without loss of generality that I ¡ C and I + C are invertible. We note that (9) is performed exactly (without rounding error) in IEEE 754 floating point arithmetic [6]. The inverse Cayley transform of A := (C + I)¡1(C ¡ I) is C = (I ¡ A)¡1(I + A). Note that since ½(C) < 1, A is well de¯ned. By Theorem 2.1 for r = 1, a lower bound on ½IR(A) yields a necessary condition for C 2 P, and an upper bound yields a su±cient condition for the P -property. This implies the following. ¡1 Theorem 3.1. For C 2 Mn(IR) not having ¡1 as an eigenvalue de¯ne A := (C + I) (C ¡ I). Then 1=2 (i) C 2 P ) max jAij Ajij < 1. i;j ¡1 (ii) kD ADk2 < 1 for some diagonal D ) C 2 P. 1=2 IR Proof. Part (i) follows by max jAijAjij · ½ (A) [17, Lemma 5.1] and Theorem 2.1. Part (ii) follows for i;j a maximizing S in (1) by IR ¡1 ¡1 ½ (A) · ½(SA) = ½(SD AD) · kD ADk2: The quantity (10) inf kD¡1ADk D is a well known upper bound for the structured singular value [3]. It can be computed e±ciently [22] using ¡D D the fact that ke Ae k2 is a convex function in the Dii [19]. Next we show that the su±cient condition (ii) in Theorem 3.1 is superior to certain other conditions for P -property. 3 Theorem 3.2. Let C 2 Mn(IR). Then T ¡1 (i) C + C positive de¯nite implies that there exists A := (C + I) (C ¡ I) and kAk2 < 1. (ii) C diagonally dominant with all diagonal elements positive implies that there exists A := (C + ¡1 ¡1 I) (C ¡ I) and inf kD ADk2 < 1, where the in¯mum is taken over all positive diagonal matrices.