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Characteristic Equation

The equation which is solved to find a Matrix's Eigenvalues, also called the Characteristic Polynomial. Given a $2\times 2$ system of equations with Matrix

{\hbox{\sf M}}\equiv \left[{\matrix{a & b\cr c & d\cr}}\right],
\end{displaymath} (1)

the Matrix Equation is
\left[{\matrix{a & b\cr c & d\cr}}\right]\left[{\matrix{x\cr y\cr}}\right] = t\left[{\matrix{x\cr y\cr}}\right],
\end{displaymath} (2)

which can be rewritten
\left[{\matrix{a-t & b\cr c & d-t\cr}}\right]\left[{\matrix{x\cr y\cr}}\right] = t\left[{\matrix{0\cr 0\cr}}\right].
\end{displaymath} (3)

M can have no Matrix Inverse, since otherwise
\left[{\matrix{x\cr y\cr}}\right]={\hbox{\sf M}}^{-1}\left[{\matrix{0\cr 0\cr}}\right] = \left[{\matrix{0\cr 0\cr}}\right],
\end{displaymath} (4)

which contradicts our ability to pick arbitrary $x$ and $y$. Therefore, M has no inverse, so its Determinant is 0. This gives the characteristic equation
\left\vert\matrix{a-t & b\cr c & d-t\cr}\right\vert = 0,
\end{displaymath} (5)

where $\vert{\hbox{\sf A}}\vert$ denotes the Determinant of A. For a general $k\times k$ Matrix
\left[{\matrix{a_{11} & a_{12} & \ldots & a_{1k}\cr
a_{21} ...
...ts & \vdots\cr
a_{k1} & a_{k2} & \ldots & a_{kk}\cr}}\right],
\end{displaymath} (6)

the characteristic equation is
\left\vert\matrix{a_{11}-t & a_{12} & \ldots & a_{1k}\cr
a_{k1} & a_{k2} & \ldots & a_{kk}-t\cr}\right\vert=0.
\end{displaymath} (7)

See also Ballieu's Theorem, Cayley-Hamilton Theorem, Parodi's Theorem, Routh-Hurwitz Theorem


Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1117-1119, 1979.

© 1996-9 Eric W. Weisstein