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Cayley-Hamilton Theorem

Given


\begin{displaymath}
\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,
\end{displaymath} (1)

then
\begin{displaymath}
{\hbox{\sf A}}^m+c_{m-1}{\hbox{\sf A}}^{m-1}+\ldots+c_0{\hbox{\sf I}}={\hbox{\sf0}},
\end{displaymath} (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


$\displaystyle \left\vert{\begin{array}{cc}a-x & b\\  c & d-x\end{array}}\right\vert$ $\textstyle =$ $\displaystyle (a-x)(d-x)-bc$  
  $\textstyle =$ $\displaystyle x^2-(a+d)x+(ad-bc)\equiv x^2+c_1x+c_2$ (3)


$\displaystyle {\hbox{\sf A}}$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}a & b\\  c & d\end{array}\right]$ (4)
$\displaystyle {\hbox{\sf A}}^2$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}a & b\\  c & d\end{array}\right]\left[\begin{array}{cc}a & b\\  c & d\end{array}\right]$  
  $\textstyle =$ $\displaystyle \left[\begin{array}{cc}a^2+bc & ab+bd\\  ac+cd & bc+d^2\end{array}\right]$ (5)
$\displaystyle -(a+d){\hbox{\sf A}}$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}-a^2-ad & -ab-bd\\  -ac-dc & -ad-d^2\end{array}\right]$ (6)
$\displaystyle (ad-bc){\hbox{\sf I}}$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}ad-bc & 0\\  0 & ad-bc\end{array}\right],$ (7)

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




© 1996-9 Eric W. Weisstein
1999-05-26