The characteristic equation is the equation which is solved to find a matrix's eigenvalues, also called the characteristic polynomial. For a general matrix , the characteristic equation in variable is defined by

where is the identity matrix and is the determinant of the matrix . Writing out explicitly gives

so the characteristic equation is given by

The solutions of the characteristic equation are called eigenvalues, and are extremely important in the analysis of many problems in mathematics and physics. The polynomial left-hand side of the characteristic equation is known as the characteristic polynomial.

See also

Ballieu's Theorem, Cayley-Hamilton Theorem, Characteristic Polynomial, Diagonal Matrix, Eigenvalue, Parodi's Theorem, Routh-Hurwitz Theorem

Explore with Wolfram|Alpha

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1117-1119, 2000.

Referenced on Wolfram|Alpha

Characteristic Equation

Cite this as:

Weisstein, Eric W. "Characteristic Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CharacteristicEquation.html

Subject classifications