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where  and  is an  matrix with real entries, has a constant solution

where  and  is an  matrix with real entries, has a constant solution

(In a different language, the origin  is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as  ("in the future") if and only if for all eigenvalues  of , . Similarly, it is asymptotically stable as  ("in the past") if and only if for all eigenvalues  of , . If there exists an eigenvalue  of  with  then the solution is unstable for .

(In a different language, the origin  is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as  ("in the future") if and only if for all eigenvalues  of , . Similarly, it is asymptotically stable as  ("in the past") if and only if for all eigenvalues  of , . If there exists an eigenvalue  of  with  then the solution is unstable for .

(在另一种语言中，它的起源是相应平衡点的动力系统。)这个解是渐近稳定的(“在未来”)当且仅当所有特征值为，。类似地，它是渐近稳定的(“在过去”)当且仅当所有特征值为，。如果存在与的特征值，则解是不稳定的。

(在另一种语言中，原点是该动力系统的平衡点。)这个解是随着{{Math|''t'' → ∞}}（未来）是渐近稳定的当且仅当对于{{Math|''A''}}的所有特征值{{Math|''λ''}}有{{Math|[[Real part|Re]](''λ'') < 0}}。类似地，它随着{{Math|''t'' → -∞}}（过去）是渐近稳定的当且仅当对于{{Math|''A''}}的所有特征值{{Math|''λ''}}有{{Math|[[Real part|Re]](''λ'') > 0}}。如果存在一个{{Math|''A''}}的特征值{{Math|''λ''}}使得{{Math|[[Real part|Re]](''λ'') > 0}}，则该解在{{Math|''t'' → ∞}}时是不稳定的。

Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots.

Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots.

为了判定线性系统原点的稳定性，劳斯-赫尔维茨稳定性判据推动了这一结果在实践中的应用。矩阵的特征值是其特征多项式的根。如果所有根的实部都是严格负的，那么一个具有实系数的单变量多项式称为赫尔维茨多项式。劳斯-赫尔维茨定理通过避免计算根的算法暗示了赫尔维茨多项式的角色塑造。

为了判定线性系统原点的稳定性，劳斯-赫尔维茨稳定性判据推动了这一结果在实践中的应用。矩阵的特征值是其特征多项式的根。如果所有根的实部都是严格负的，那么一个具有实系数的单变量多项式称为赫尔维茨多项式。劳斯-赫尔维茨定理通过避免计算根的算法暗示了赫尔维茨多项式的特征。

=== Non-linear autonomous systems ===

=== Non-linear autonomous systems ===