Say that we have a square matrix of order $N \times N$ and is NOT a diagonal matrix. Now can this matrix satisfy a polynomial of degree $N$ other than its own characteristic equation?
Does a square matrix satisfy only its characteristic equation
cayley-hamiltonmatrix equations
Best Answer
For sure, for example let $M$ be any $n \times n$ matrix such that $M \neq 0$ but $M^2 = 0$, for example we could take $M$ to be a matrix full of zeros, with a single 1 in the top-right spot. The characteristic equation of $M$ is $x^n$, but $M$ satisfies many more polynomials of degree $n$ as long as $n > 2$, for example it satisfies $M^2(M + 1) = 0$.
In general, this happens whenever the minimal polynomial of a matrix is a lower degree than the characteristic polynomial.