Cayley-Hamilton Theorem

My final post (so far) on matrix conjugation is the most useful of matrix theories and one of my personal favourites.

"In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation."

(http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem )

This means that once you have the characteristic polynomial of a matrix M (for a 3x3 matrix in this example):

P(x) = a₀ + a₁x + a₂x² + x³= 0

then replacing x with M:

P(M) = a₀I + a₁M + a₂M² + M³= 0

will also be true.

Notice that we've replaced the constant term a₀ with a₀I, guaranteeing that this will give us a square matrix.