Topics: Algebra - Diagonalisation of a Linear Transformation

This theorem assures us that a linear transformation (with a vector space on with finite dimension) nullifies its own characteristic polynomial.


Let be a linear transformation, and be its characteristic polynomial.



Let be the associated matrix of , and be its characteristic polynomial.

Then, .

Theorem applications

Finding the inverse of

We’ll find the inverse of .

1. Find the characteristic polynomial

We then substitute for , and multiply the constant term by the identity matrix:

(it’s equal to since the theorem tells us that the characteristic polynomial is nullified).


We then multiply this by :

Operating with the actual matrix, we get:

Voilà. We got that .