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Characteristic Equation


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

 det(A-lambdaI)=0,
(1)

where I is the identity matrix and det(B) is the determinant of the matrix B. Writing A out explicitly gives

 A=[a_(11) a_(12) ... a_(1k); a_(21) a_(22) ... a_(2k); | | ... |; a_(k1) a_(k2) ... a_(kk)],
(2)

so the characteristic equation is given by

 |a_(11)-lambda a_(12) ... a_(1k); a_(21) a_(22)-lambda ... a_(2k); | | ... |; a_(k1) a_(k2) ... a_(kk)-lambda|=0
(3)

The solutions lambda 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

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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 Web Resource. https://mathworld.wolfram.com/CharacteristicEquation.html

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