Cayley-Hamilton Theorem

Given

$$\left\vert{\matrix{a_{11}-x & a_{12} & \cdots & a_{1m}\cr a_... ...a_{mm}-x\cr}}\right\vert\hfill =x^m+c_{m-1}x^{m-1}+\ldots+c_0,$$ (1)

then

$${\hbox{\sf A}}^m+c_{m-1}{\hbox{\sf A}}^{m-1}+\ldots+c_0{\hbox{\sf I}}={\hbox{\sf0}},$$ (2)

where I is the Identity Matrix. Cayley verified this identity for $m=2$ and 3 and postulated that it was true for all $m$. For $m=2$, direct verification gives

$$\left\vert{\begin{array}{cc}a-x & b\\ c & d-x\end{array}}\right\vert = (a-x)(d-x)-bc$$

$$= x^2-(a+d)x+(ad-bc)\equiv x^2+c_1x+c_2$$ (3)

$${\hbox{\sf A}} = \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]$$ (4)

$${\hbox{\sf A}}^2 = \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]\left[\begin{array}{cc}a & b\\ c & d\end{array}\right]$$

$$= \left[\begin{array}{cc}a^2+bc & ab+bd\\ ac+cd & bc+d^2\end{array}\right]$$ (5)

$$-(a+d){\hbox{\sf A}} = \left[\begin{array}{cc}-a^2-ad & -ab-bd\\ -ac-dc & -ad-d^2\end{array}\right]$$ (6)

$$(ad-bc){\hbox{\sf I}} = \left[\begin{array}{cc}ad-bc & 0\\ 0 & ad-bc\end{array}\right],$$ (7)

so

$${\hbox{\sf A}}^2-(a+d){\hbox{\sf A}}+(ad-bc){\hbox{\sf I}}=\left[{\matrix{0 & 0\cr 0 & 0\cr}}\right].$$ (8)

The Cayley-Hamilton theorem states that a $n\times n$ Matrix A is annihilated by its Characteristic Polynomial ${\rm det}(x{\hbox{\sf I}}-{\hbox{\sf A}})$, which is monic of degree $n$.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1117, 1979.

Segercrantz, J. ``Improving the Cayley-Hamilton Equation for Low-Rank Transformations.'' Amer. Math. Monthly 99, 42-44, 1992.