Sign patterns requiring a unique inertia Jephian C.-H. Lin - PowerPoint PPT Presentation

Sign patterns requiring a unique inertia Jephian C.-H. Lin 林 晉 宏 Department of Applied Mathematics, National Sun Yat-sen University May 25, 2019 7th TWSIAM Annual Meeting, Hsinchu, Taiwan Sign patterns requiring a unique inertia 1/22 NSYSU

Joint work with Pauline van den Driessche D. Dale Olesky University of Victoria University of Victoria Sign patterns requiring a unique inertia 2/22 NSYSU

Sign pattern ◮ A sign pattern is a matrix whose entries are in { + , − , 0 } . ◮ The qualitative class of a sign pattern P = � � is the family p i , j � � of matrices A = a i , j such that sign( a i , j ) = p i , j . �� + �� � 1 � � 5 � + 0 2 0 3 0 Q ∋ , , . . . 0 − + 0 − 3 0 . 5 0 − 1 π Sign patterns requiring a unique inertia 3/22 NSYSU

Require and allow ◮ Let P be a sign pattern. ◮ Let R be a property of a matrix. E.g., being invertible, being nilpotent, etc. ◮ P requires property R if every matrix in Q ( P ) has property R . ◮ P allows property R if at least a matrix in Q ( P ) has property R .   + 0 0  requires a positive determinant. − + 0  0 − + Sign patterns requiring a unique inertia 4/22 NSYSU

Require and allow ◮ Let P be a sign pattern. ◮ Let R be a property of a matrix. E.g., being invertible, being nilpotent, etc. ◮ P requires property R if every matrix in Q ( P ) has property R . ◮ P allows property R if at least a matrix in Q ( P ) has property R .  + 0 −   allows a positive determinant. − + 0  0 − + Sign patterns requiring a unique inertia 4/22 NSYSU

Inertia Let A be a matrix. ◮ n + ( A ) = number of eigenvalues with positive real part. ◮ n − ( A ) = number of eigenvalues with negative real part. ◮ n 0 ( A ) = number of eigenvalues with zero real part. ◮ n z ( A ) = number of eigenvalues that equals zero. The inertia of A is the triple ( n + ( A ) , n − ( A ) , n 0 ( A )) . yi x n − = 3 n + = 1 n 0 = 2 Sign patterns requiring a unique inertia 5/22 NSYSU

Question: ◮ Which sign pattern requires a unique inertia? Outlines: ◮ Motivations from dynamical systems ◮ Sign patterns requiring a unique inertia Sign patterns requiring a unique inertia 6/22 NSYSU

+ + − − Predator-Prey Model dx dt = α x − β xy dy dt = δ xy − γ y (pictures from Wikipedia) Sign patterns requiring a unique inertia 7/22 NSYSU

General form � � Let P = be a sign pattern. Let x i , j be variables for i , j ∈ [ n ] . p i , j The general form of P is a variable matrix X with  if p i , j = +; x i , j   ( X ) i , j = − x i , j if p i , j = − ;  0 if p i , j = 0 .  � x 1 , 1 � + � � + x 1 , 2 P = X = − − − x 2 , 1 − x 2 , 2 Write det( zI − X ) = S 0 z n − S 1 z n − 1 + S 2 z n − 2 + · · · + ( − 1 ) n S n . Then each S k is a multivariate polynomial in x i , j ’s. Sign patterns requiring a unique inertia 8/22 NSYSU

Sign of a polynomial ◮ Let p be a polynomial. ◮ p can be expanded into a linear combination of non-repeated monomials.  0 if all coefficients = 0 ;     + if all nonzero coefficients > 0 and sign( p ) � = 0 ;  sign( p ) = − if all nonzero coefficients < 0 and sign( p ) � = 0 ;     # otherwise.  Sign patterns requiring a unique inertia 9/22 NSYSU

Minor sequence ◮ Let X be the general form a sign pattern P . The minor sequence of P is s 0 , s 1 , . . . , s n , where s k = sign( S k ) . Theorem (JL, Olesky, and van den Driessche 2018) If s n = # , then P does not require a unique inertia. When P is a 2 × 2 sign pattern, P require a unique inertia if and only if s 2 � = # . � 0 � � 0 � � 0 � � 0 � + + − − − 0 + 0 − + + + [+ , 0 , +] [+ , 0 , − ] [+ , + , − ] [+ , + , +] � − � � − � � + � � + � − − − + − + + + − + − + [+ , # , − ] [+ , # , #] [+ , + , #] [+ , + , +] Sign patterns requiring a unique inertia 10/22 NSYSU

Equivalence conditions Theorem (JL, Olesky, and van den Driessche 2018) Let P be a sign pattern. The following are equivalent: ◮ P requires a unique inertia. ◮ P requires a fixed n 0 . ◮ P requires a fixed n z and a fixed number of nonzero pure imaginary eigenvalues. yi x Sign patterns requiring a unique inertia 11/22 NSYSU

Number of nonzero pure imaginary roots Substitute z by ti (with t � = 0): p ( z ) = x 5 + x 4 + 6 x 3 + 2 x 2 + 9 x − 3 = ( t 4 − 6 t 2 + 9 ) ti + ( t 4 − 2 t 2 − 3 ) odd part = x 2 − 6 x + 9 even part = x 2 − 2 x − 3 # of nonzero pure imaginary roots = 2 · # of common positive roots of the odd and the even parts For det( zI − X ) , odd part : S 0 , − S 2 , S 4 , . . . even part : S 1 , − S 3 , S 5 , . . . Sign patterns requiring a unique inertia 12/22 NSYSU

Number of nonzero pure imaginary roots Substitute z by ti (with t � = 0): p ( z ) = x 5 + x 4 + 6 x 3 + 2 x 2 + 9 x − 3 = ( t 4 − 6 t 2 + 9 ) ti + ( t 4 − 2 t 2 − 3 ) odd part = x 2 − 6 x + 9 even part = x 2 − 2 x − 3 # of nonzero pure imaginary roots = 2 · # of common positive roots of the odd and the even parts For det( zI − X ) , odd part : S 0 , − S 2 , S 4 , . . . even part : S 1 , − S 3 , S 5 , . . . Sign patterns requiring a unique inertia 12/22 NSYSU

Descartes’ rule of signs Theorem (Descartes’ rule of signs) Suppose p ( x ) � = 0 is a polynomial whose coefficients has t sign changes (ignoring the zeros). Then p ( x ) has t − 2 k positive roots for some k ≥ 0 . For example ◮ x 2 − 6 x + 9 has 2 or 0 positive roots, and ◮ x 2 + 0 x − 4 has 1 positive root. [Key: No sign changes, no positive roots!] Sign patterns requiring a unique inertia 13/22 NSYSU

Descartes’ rule of signs Theorem (Descartes’ rule of signs) Suppose p ( x ) � = 0 is a polynomial whose coefficients has t sign changes (ignoring the zeros). Then p ( x ) has t − 2 k positive roots for some k ≥ 0 . For example ◮ x 2 − 6 x + 9 has 2 or 0 positive roots, and ◮ x 2 + 0 x − 4 has 1 positive root. [Key: No sign changes, no positive roots!] Sign patterns requiring a unique inertia 13/22 NSYSU

Resultant k = 0 c k x ℓ − k and p 2 ( x ) = � m Let p 1 ( x ) = � ℓ k = 0 d k x m − k . The Sylvester matrix of p 1 and p 2 is an ( m + ℓ ) × ( m + ℓ ) matrix  c 0 d 0  c 1 c 0 d 1 d 0   ... ...   c 2 c 1 d 2 d 1     . . ... ...   . . . c 0 . d 0     . .   . . S ( p 1 , p 2 ) = . . c 1 . d 1     c ℓ d m     . .  . .  c ℓ . d m .     ... ...     c ℓ d m Sign patterns requiring a unique inertia 14/22 NSYSU

The resultant of p 1 and p 2 is Res( p 1 , p 2 ) = det( S ( p 1 , p 2 )) . Theorem Res( p 1 , p 2 ) = 0 if and only if p 1 and p 2 have a common factor. Suppose P is a sign pattern with general form X . ◮ Res( P ) = Res( even part , odd part ) with the two parts from det( zI − X ) . Sign patterns requiring a unique inertia 15/22 NSYSU

The resultant of p 1 and p 2 is Res( p 1 , p 2 ) = det( S ( p 1 , p 2 )) . Theorem Res( p 1 , p 2 ) = 0 if and only if p 1 and p 2 have a common factor. Suppose P is a sign pattern with general form X . ◮ Res( P ) = Res( even part , odd part ) with the two parts from det( zI − X ) . Sign patterns requiring a unique inertia 15/22 NSYSU

 0 0  x 1 , 2  , − x 2 , 1 0 − x 2 , 3  0 − x 3 , 2 x 3 , 3 S 0 ( P ) = 1 S 2 ( P ) = x 1 , 2 x 2 , 1 − x 2 , 3 x 3 , 2 S 1 ( P ) = x 3 , 3 S 3 ( P ) = x 1 , 2 x 2 , 1 x 3 , 3 Res( P ) = x 3 , 3 ( x 1 , 2 x 2 , 1 − x 2 , 3 x 3 , 2 ) − x 1 , 2 x 2 , 1 x 3 , 3 = x 3 , 3 x 1 , 2 x 2 , 1 − x 3 , 3 x 2 , 3 x 3 , 2 − x 1 , 2 x 2 , 1 x 3 , 3 = x 3 , 3 x 2 , 3 x 3 , 2 . sign(Res( P )) = + = ⇒ never has common positive roots So, P does not allow any nonzero pure imaginary eigenvalues. Sign patterns requiring a unique inertia 16/22 NSYSU

Exceptional, not exceptional A 3 × 3 sign pattern P is in E if its minor sequence is [+ , # , # , +] or [+ , # , # , − ] Sign patterns requiring a unique inertia 17/22 NSYSU

3 × 3 sign patterns not in E Theorem (JL, Olesky, and van den Driessche 2018) Let P be a 3 × 3 irreducible sign pattern that is not in E . Then P requires a unique inertia if and only if 1. s k 0 ∈ { + , −} and s k = 0 for all k > k 0 (fixed n z ), and 2. At least one of the following holds: (fixed n 0 − n z = 0 ) 2.1 s 2 = − . (no sign changes in even part) 2.2 s 1 , s 3 ∈ { + , − , 0 } and s 1 � = s 3 . (no sign changes in odd part) 2.3 Res( P ) has a fixed sign. Sign patterns requiring a unique inertia 18/22 NSYSU

Embedded T 2 � + � + T 2 = allows two inertias ( 2 , 0 , 0 ) and ( 0 , 2 , 0 ) − −  + + 0   allows two inertias ( 2 , 0 , 1 ) and ( 0 , 2 , 1 ) − − 0  0 0 0 Lemma (JL, Olesky, and van den Driessche 2018) If P is a 3 × 3 sign pattern with T 2 (or T ⊤ 2 ) embedded in P as a principal subpattern, then P does not require a unique inertia.  0 0 +   has minor sequence [+ , # , # , +] − + +  0 − − Sign patterns requiring a unique inertia 19/22 NSYSU