# The Stability Criteria with Compound Matrices

Authors: Yazhuo Zhang; Baodong Zheng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The bifurcation problem is one of the most important subjects in dynamical systems. Motivated by M. Li et al. who used compound matrices to judge the stability of matrices and the existence of Hopf bifurcations in continuous dynamical systems, we obtained some effective methods to judge the Schur stability of matrices on the base of the spectral property of compound matrices, which can be used to judge the asymptotical stability and the existence of Hopf bifurcations of discrete dynamical systems.


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## Body

## 1. Introduction

The stability of matrices is intimately related to the stability of stationary solutions of various kinds in the theory and applications of dynamic systems [1]. Let A be an n×n matrix, and let σ(A) be its spectrum. The stability modulus of A is s(A)=max{Reλ:λ∈σ(A)}, and A is said to be stable if s(A)<0. The asymptotic stability of the equilibrium point of a continuous dynamical system is closely related to the stability of the Jacobian matrix of the system at the equilibrium point; it is necessary that stability modulus is less than or equal to zero and it is sufficient that the Jacobian matrix is stable for the equilibrium point to be asymptotically stable. The stability of a matrix is related to the Routh-Hurwitz problem [2] on the number of zeros of a polynomial that have negative real parts. Li and Wang [3] have given a necessary and sufficient condition for the stability of an n×n matrix with real entries using the additive compound matrix. Accordingly, some criteria on the stability of the continuous dynamical systems are obtained.The spectral radius ofA is ρ(A)=max{|λ|:λ∈σ(A)}, and A is said to be Schur stable if ρ(A)<1. On the other hand, the asymptotic stability of a fixed point of a discrete dynamical system is closely related to the Schur stability of the linearized matrix of the system [4]; it is necessary that spectral radius of the linearized matrix is less than or equal to one and it is sufficient that the linearized matrix is Schur stable for the fixed point to be asymptotically stable. Corresponding to the Routh-Hurwitz criterion, the Jury criterion was derived by Jury in 1974. And Zheng et al. [5] obtained the extended Jury criterion on the coefficients of the characteristic polynomial of the matrix.By the spectral property of the compound matrix, the compound matrix is an excellent tool to research the stability of matrices [6, 7]. In the present paper, using the additive compound matrix and the Lozinskii measure, some conclusions about the Schur stability of matrices and the stability  of  the discrete dynamical system are derived. Using the multiplicative compound matrix, we demonstrate how to judge the Schur stability of a matrix; at the same time we give the conditions that there exist no Hopf bifurcations in the discrete dynamical systems and the conditions for the fixed point to be asymptotically stable in the discrete dynamical systems.We outline in the next section the preliminaries about the compound matrix and the Lozinskii measure and so on. In the Sections3 and 4, the main results are demonstrated.


## 2. Preliminaries

LetMn(T) be the linear space of n×n matrices with entries in T, where T is the field of real numbers R or complex numbers C. An A∈Mn(T) will be identified with the linear operator on Tn that it represents. Let ∧ denote the exterior product in Tn, and let k, 1≤k≤n, be an integer.Definition 1 (see [6]).
With respect to the canonical basis in thekth exterior product space ∧kTn, the kth additive compound matrix A[k] of A and the kth multiplicative compound matrix A(k) of A are, respectively, linear operators on ∧kTn whose definitions on a decomposable element u1∧⋯∧uk are
(1)A[k](u1∧⋯∧uk)=∑j=1k‍u1∧⋯∧Auj∧⋯∧uk,∀u1,…,uk∈Tn,A(k)(u1∧⋯∧uk)=Au1∧⋯∧Auk,∀u1,…,uk∈Tn.
Definition over the whole∧kTn is done by linear extension. The entries of A[k] and A(k) are linear relations of those of A.LetA=(aij),   and let  N=Cnk. For any integer i=1,2,…,N, let i=(i1,…,ik) be the ith member in the lexicographic ordering of integer k-tuples such that 1≤i1<⋯<ik≤n. Then the following conclusions are concluded.Proposition 2 (see [6]).
We get the following:(2)A(k)=(a(i)(j))N×N.
And letZ=A[k]=(zij); then
(3)zij={ai1i1+⋯+aikik,if(i)=(j),(-1)r+sajris,ifexactlyoneentryisof(i)doesnotoccurin(j)andjrdoesnotoccurin(i),0,if(i)differsfrom(j)intwoormoreentries.As special cases, we haveA[1]=A and A[n]=tr(A). The spectral property is as follows.Proposition 3 (see [6]).
Letσ(A)={λi:i=1,…,n}: then(1)
σ(A[k])={λi1+⋯+λik:1≤i1<⋯<ik≤n},(2)
σ(A(k))={λi1⋯λik:1≤i1<⋯<ik≤n}.Pertinent to the purpose of the present paper is the norm property of the compound matrix.The singular valuesσ1,…,σn of an n×n matrix B are the eigenvalues of the symmetric matrix B*B, and let
(4)σ1≥⋯≥σn≥0.Proposition 4 (see [7] (minimax-rule)).
The following is established:(5)σj=maxV⊂RmdimV=jminu∈V∥u∥=1∥Bu∥=minW⊂RmdimW=m-j+1maxu∈W∥u∥=1∥Bu∥,j=1,…,n, where ∥·∥ are the matrix norm induced from the Euclidean norm of Rn.Corollary 5 (see [7]).
The following is established:σ1=∥B∥.Definition 6 (see [3]).
Let|·| denote a vector norm in Tn and the operator norm it induces in Mn(T). For A∈Mn(T), the Lozinskii measure μ on Mn(T) with respect to |·| is defined by
(6)μ(A)=limh→0+|I+hA|-1h.The following are the stability criteria on matrices given by Li and Wang using the compound matrix and Lozinskii measure.Proposition 7 (see [3]).
We get the following:s(A)<0⇔s(A[2])<0 and (-1)ndet(A)>0.Proposition 8 (see [3]).
If(-1)ndet(A)>0, then A is stable ⇔ there exists a Lozinskii measure μ on MN(R) such that μ(A[2])<0,N=Cn2.Definition 9 (see [3]).
Letα↦A(α)∈Mn(R) be a continuous function on (a,b). α0∈(a,b) is called a Hopf bifurcation point of A(α), if, when α<α0, A(α) is stable. When α>α0, there exist a pair of plural eigenvalues Reλ(α)±iImλ(α) such that Reλ(α)>0, and the other eigenvalues have nonzero real parts; here Imλ(α)≠0.Proposition 10 (see [3]).
Consider the system(7)x′=f(x,α),α∈(a,b),x∈D,f∈C1(Rn×(a,b),Rn),
where f,∂f/∂x∈C(Rn×(a,b),Rn) and D is an open set in Rn. If, for all α∈(a,b), f(x,α)=0 have solutions (x-(α),α), then x-(α) is called the equilibrium of the system (7). Let J(α)=(∂f/∂x)(x-(α),α), and if α=α0 is the Hopf bifurcation point of J(α), then the Hopf bifurcation of system (7) occurs when α=α0.By the theorem, Li and Wang have given the methods to judge the existence of the Hopf bifurcation.Proposition 11 (see [3]).
If there exists some Lozinskii measureμ such that
(8)μ(J[2](α))≤0,α∈(a,b),
then the system (7) has no Hopf bifurcations from x-(α) in (a,b). Moreover, if
(9)(-1)ndet(J(α))>0,α∈(a,b),
then x-(α) is an asymptotically stable equilibrium of the system (7) for α∈(a,b).


## 3. The Schur Stability Criteria of Matrices Using the Additive Compound Matrix

Consider theCr(r≥1) map:
(10)x⟼g(x),x∈Rn.
If (10) has a fixed point x=x-, that is,  x-=g(x-), then the linear map corresponding to (10) is
(11)y⟼Ay,y∈Rn,
where A≡Dg(x-), the Jacobian matrix of g at x-.Similar to the continuous dynamical system, the stability of a discrete dynamical system can be judged by the Schur stability of the matrix.Lemma 12 (see [4]).
If the matrixA of the system (11) is Schur stable, then the fixed point x- of the system (11) is asymptotically stable.It is generally well known that it is a sufficient condition for the Hopf bifurcation to occur in the system (10) that the matrix A has a pair of conjugated eigenvalues whose spectral radius is equal to 1 (not equal to ±1), and the spectral radius of the rest of the  eigenvalues is less than 1. In addition, it is a sufficient condition for the k-codimension Hopf bifurcation to occur in the system (10) that the matrix A has k pairs of conjugated of the eigenvalues whose spectral radius is equal to 1 (not equal to ±1), and the spectral radius of the rest eigenvalues is less than 1.Thus, it is very necessary to investigate the following three questions:(1)
the conditions for the spectral radius of all the eigenvalues of the matrixA to be less than 1;(2)
the conditions for the spectral radius of a pair of the eigenvalues of the matrixA to be equal to 1 (not equal to ±1) and the spectral radius of the rest of the eigenvalues of the matrix A to be less than 1;(3)
the conditions for the spectral radius ofk pairs of the eigenvalues of the matrix A to be equal to 1 (not equal to ±1) and the spectral radius of the rest of the eigenvalues of the matrix A to be less than 1.For the characteristic polynomial of the matrixA:
(12)fn(z)=det(zIn-A)=zn+a1zn-1+⋯+an,
we consider the operation
(13)z=ω+1ω-1,
and (12) is changed into
(14)fn=ωn+b1ωn-1+⋯+bn.By operation (13), the  inner  of the unit circle and the unit circle are mapped to left plane and the imaginary axis of the ω complex plane. And the questions about the eigenvalues of the matrices of the discrete dynamical systems are transformed into the questions about the eigenvalues of the matrices of the continuous dynamical systems. Therefore the following conclusions are obtained.Theorem 13.
If(-1)ndet(I+(2/det(A-I))(A-I)*)>0, then A is Schur stable ⇔ there exists some Lozinskii measure μ such that
(15)μ(I+2det(A-I)(A-I)*)[2]<0.Proof.
The eigenvalues ofI+(2/det(A-I))(A-I)* are
(16)1+2det(A-I)·det(A-I)λ-1=1+2λ-1=λ+1λ-1,
where λ is the eigenvalue of the matrix A.
Therefore by (13),
(17)ρ(A)<1⟺s(I+2det(A-I)(A-I)*)<0.
By Proposition 8, the conclusions are obtained.Theorem 14.
The spectral radius of a pair of conjugated eigenvalues of the matrixA is equal to 1 (not equal to ±1) and the spectral radius of the rest eigenvalues is less than 1⇔ there exists a Lozinskii measure μ such that μ(I+(2/det(A-I))(A-I)*)[3]<0, (-1)ndet(I+(2/det(A-I))(A-I)*)>0, s(I+(2/det(A-I))(A-I)*)[2]=0.Proof.
⇒ It is obvious.
⇐ Let
(18)I+2det(A-I)(A-I)*=A~.
By s(A~)[2]=0,  A~ has at most one eigenvalue that its real part is more than zero, and the rest of the eigenvalues have real parts less than zero. Without loss of generality, we assume that λn>0 and λn is a real number. But
(19)sign(detA~)=(-1)n-1.
It contradicts
(20)(-1)ndetA~>0.
And by
(21)μ(A~)[3]<0⟺s(A~)[3]<0,
it can be known that A~ has only one pair of eigenvalues whose real parts are zero. Thus A~ has only one pair of eigenvalues whose real parts are zero, and the real parts of the rest of the eigenvalues are less than zero. And the conclusions are established.Consider theCr(r≥1) map:
(22)x⟼g(x,α),x∈Rn,α∈(a,b).
If (22) has the fixed point x=x-(α), that is, x-(α)=g(x-,α), then the linear map corresponding to (22) is
(23)y⟼A(α)y,y∈Rn,α∈(a,b),
where A≡Dg(x-(α),α), the Jacobian matrix of g at x-(α).Definition 15.
α
0
∈
(
a
,
b
) is called the Hopf bifurcation point if, when α<α0,  A(α) is Schur stable; when α>α0, there exist a pair of complex eigenvalues whose spectral radius is more than 1, and the rest of the eigenvalues are not 1 or -1.Therefore the following theorem is obtained.Theorem 16.
If there exists some Lozinskii measureμ such that μ(I+(2/det(J(α)-I))(J(α)-I)*)[2]≤0, α∈(a,b), then the system (22) has no Hopf bifurcation points about its fixed point x-(α). And if (-1)ndet(I+(2/det(J(α)-I))(J(α)-I)*)>0,α∈(a,b), then x-(α) is an asymptotically stable fixed point of the system (22), where g,∂g/∂x∈C(Rn×(a,b),Rn), J(α)=(∂g/∂x)(x-(α),α), and x-(α) is the fixed point of the system (22).


## 4. The Schur Stability Criteria of Matrices Using the Multiplicative Compound Matrix

To research the Schur stability of the matrix and the stability of the discrete dynamical system,  we find that the measure whose function is similar to that of the Lozinskii measure in the research of  continuous dynamical systems is the norm of the matrix.LetMn(C) denote all the n×n matrices in the complex field.Lemma 17.
LetA∈Mn(C); then ρ(A)=inf{|A|:|·| is the matrix norm in  Mn(C)}.Thus we have the following corollary.Corollary 18.
LetA∈Mn(C); then ρ(A)<1⇔ there exists some matrix norm in Mn(C) such that |A|<1.By the spectral property of the multiplicative compound matrix, the Schur stability of the matrix can be judged by the multiplicative compound matrix.Theorem 19.
LetA∈Mn(R); then ρ(A)<1⇔ρ(A(2))<1 and det(I-A2)>0.Proof.
⇒ Because ρ(A)<1, by Proposition 3, ρ(A(2))<1. It might be as well that the n roots of A(2) are
(24)ν1,…,νs,νs+1=ν-1,…,ν2s=ν-s,ν2s+1,…,νn,
where ν2s+1,…,νn are real numbers. Then by ρ(A)<1, we have
(25)(1-νi)(1-ν-i)>0,i=1,…,s,1-νj>0,j=2s+1,…,n.
Thus
(26)det(I-A2)=∏i=1s‍(1-νi)(1-ν-i)∏j=2s+1n‍(1-νj)>0.
⇐ By ρ(A(2))<1 and Proposition 3, A has at most one eigenvalue whose spectral radius is more than 1. If ρ(A)≥1, by A∈Mn(R), it can be known that A has just one real eigenvalue λ0 whose absolute value is more than or equal to 1. Let
(27)f(x)=|xI-A|=xn-tr(A)xn-1+⋯+(-1)n|A|
be the characteristic polynomial of A.
By(28)limx→+∞f(x)=+∞,limx→-∞(-1)nf(x)=+∞,
it can be known that if λ0≥1, then
(29)det(I-A)≤0,(-1)ndet(-I-A)>0.
Thus
(30)det(I-A2)<0,
a contradiction.
Ifλ0≤-1, then
(31)(-1)ndet(-I-A)≤0,det(I-A)>0.
Thus
(32)det(I-A2)<0,
a contradiction.
Therefore the conclusions are established.By the previous conclusions and corollaries, we have the following.Corollary 20.
LetA∈Mn(R); then ρ(A)<1⇔ there exists some matrix norm in MN(R) such that |A(2)|<1 and det(I-A2)>0, where N=Cn2.By the property of the multiplicative compound matrix given in Section2, the multiplicative compound matrix can be computed by the singular value of a matrix, and thus we have the following.Corollary 21.
LetA∈Mn(R); then ρ(A)<1⇔σ1σ2<1 and det(I-A2)>0, σ1σ2 are the singular values of A.Corresponding to the stability of the fixed point and the Hopf bifurcation of the discrete dynamical systems, we have the following theorem.Theorem 22.
If there exists some matrix norm inMN(R) such that |J(2)(α)|≤1, α∈(a,b), then the system (22) has no Hopf bifurcations about its fixed point x-(α). And if det(I-A2)>0, α∈(a,b), then x-(α) is an asymptotically stable fixed point of the system (22), where g,∂g/∂x∈C(Rn×(a,b),Rn), J(α)=(∂g/∂x)(x-(α),α), and x-(α) is the fixed point of the system (22).



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