# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 12: The Cayley-Hamilton Theorem

The Cayley-Hamilton Theorem states that a linear operator $T$ satisfies (is annihilated by) its own characteristic polynomial.

That is, if $T: V \rightarrow V$ is a linear operator and

$f(t) = a_0 + a_1t + a_2t^2 + \dots + a_nt^n$

is its characteristic polynomial, then

$f(T) = a_0I + a_1T + a_2T^2 + \dots + a_nT^n = T_0$,

where $T_0$ is the zero transformation.