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  2. Characteristic equation (calculus) - Wikipedia

    en.wikipedia.org/wiki/Characteristic_equation...

    The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative.

  3. Linear recurrence with constant coefficients - Wikipedia

    en.wikipedia.org/wiki/Linear_recurrence_with...

    which is the characteristic equation of the recurrence relation. Solve for to obtain the two roots , : these roots are known as the characteristic roots or eigenvalues of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution

  4. Characteristic polynomial - Wikipedia

    en.wikipedia.org/wiki/Characteristic_polynomial

    The characteristic equation, also known as the determinantal equation, [1] [2] [3] is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory , the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix .

  5. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those ...

  6. Unit root - Wikipedia

    en.wikipedia.org/wiki/Unit_root

    In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation.

  7. Routh–Hurwitz stability criterion - Wikipedia

    en.wikipedia.org/wiki/Routh–Hurwitz_stability...

    The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions e pt of the system that are stable . Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly.

  8. Stable polynomial - Wikipedia

    en.wikipedia.org/wiki/Stable_polynomial

    In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: all its roots lie in the open left half-plane, or; all its roots lie in the open unit disk.

  9. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    We call p(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an N th-order polynomial equation in the unknown λ. This equation will have N λ distinct solutions, where 1 ≤ N λ ≤ N. The set of solutions, that is, the eigenvalues, is called the spectrum of A. [1] [2] [3]