Characteristic Polynomial

If \((\lambda, v)\) is an eigenpair of a square matrix \(A\) of order \(n\), then: \[ \begin{aligned} Av &= \lambda v\\ Av &= \lambda (Iv)\\ Av &= (\lambda I)v\\ Av - (\lambda I) v &= 0\\ (A - \lambda I)v &= 0 \end{aligned} \] Since \(v \neq 0\), \(A - \lambda I\) must have non-zero nullity. This implies that \(A - \lambda I\) is not full rank. From this it follows that \(|A - \lambda I| = 0\). We now claim that \(|A - \lambda I|\) is a polynomial in \(\lambda\) of degree \(n\). For the proof of this statement, refer Terence Tao’s notes.