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Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix is annihilated by its characteristic polynomial. For an m×m matrix A=(a_(ij)), let I be the identity matrix and write the characteristic polynomial p(x)=det(xI-A) as

p(x)=|x-a_(11) -a_(12) ... -a_(1m); -a_(21) x-a_(22) ... -a_(2m); | | ... |; -a_(m1) -a_(m2) ... x-a_(mm)|
(1)
=x^m+c_(m-1)x^(m-1)+...+c_0.
(2)

This is a monic polynomial of polynomial degree m. The theorem gives

A^m+c_(m-1)A^(m-1)+...+c_0I=0.
(3)

Cayley (1858, pp. 23-24) verified this identity for m=2 and 3 and stated it for arbitrary m without giving a general proof. For m=2, direct verification gives

|x-a -b; -c x-d|=(x-a)(x-d)-bc
(4)
=x^2-(a+d)x+(ad-bc)
(5)
=x^2+c_1x+c_0
(6)
A=[a b; c d]
(7)
A^2=[a b; c d][a b; c d]
(8)
=[a^2+bc ab+bd; ac+cd bc+d^2]
(9)
-(a+d)A=[-a^2-ad -ab-bd; -ac-dc -ad-d^2]
(10)
(ad-bc)I=[ad-bc 0; 0 ad-bc],
(11)

so

A^2-(a+d)A+(ad-bc)I=[0 0; 0 0].
(12)

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References

Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 181, 1962.Cayley, A. "A Memoir on the Theory of Matrices." Phil. Trans. Roy. Soc. London 148, 17-37, 1858. https://doi.org/10.1098/rstl.1858.0002.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1117, 2000.Segercrantz, J. "Improving the Cayley-Hamilton Equation for Low-Rank Transformations." Amer. Math. Monthly 99, 42-44, 1992.

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Cayley-Hamilton Theorem

Cite this as:

Weisstein, Eric W. "Cayley-Hamilton Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cayley-HamiltonTheorem.html

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