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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.
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 .
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 ...
In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. [1] [2] The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices. [3]
For example, if A is a multiple aI n of the identity matrix, then its minimal polynomial is X − a since the kernel of aI n − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a) n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the dimension of the space).
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial. [1] If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.
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.
If the roots of the characteristic polynomial are not distinct, and is a root of multiplicity, then () in the formula has degree . For instance, if the characteristic polynomial factors as ( x − r ) 3 {\displaystyle (x-r)^{3}} , with the same root r occurring three times, then the n {\displaystyle n} th term is of the form s n = ( a + b n + c ...