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

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

the Matrix Equation is

$$\left[{\matrix{a & b\cr c & d\cr}}\right]\left[{\matrix{x\cr y\cr}}\right] = t\left[{\matrix{x\cr y\cr}}\right],$$ (2)

which can be rewritten

$$\left[{\matrix{a-t & b\cr c & d-t\cr}}\right]\left[{\matrix{x\cr y\cr}}\right] = t\left[{\matrix{0\cr 0\cr}}\right].$$ (3)

M can have no Matrix Inverse, since otherwise

$$\left[{\matrix{x\cr y\cr}}\right]={\hbox{\sf M}}^{-1}\left[{\matrix{0\cr 0\cr}}\right] = \left[{\matrix{0\cr 0\cr}}\right],$$ (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

$$\left\vert\matrix{a-t & b\cr c & d-t\cr}\right\vert = 0,$$ (5)

where $\vert{\hbox{\sf A}}\vert$ denotes the Determinant of A. For a general $k\times k$ Matrix

$$\left[{\matrix{a_{11} & a_{12} & \ldots & a_{1k}\cr a_{21} ... ...ts & \vdots\cr a_{k1} & a_{k2} & \ldots & a_{kk}\cr}}\right],$$ (6)

the characteristic equation is

$$\left\vert\matrix{a_{11}-t & a_{12} & \ldots & a_{1k}\cr a_... ...dots\cr a_{k1} & a_{k2} & \ldots & a_{kk}-t\cr}\right\vert=0.$$ (7)

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

References

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