# Does the Minimum Polynomial of a Matrix depend continuously on the entries of the matrix?

We know that the coefficients of the characteristic polynomial of a matrix depend continuously on the entires of the matrix (since they are polynomials in the entires of a matrix).

The coefficients of minimum polynomial are not simply polynomials in the entries of a matrix, but can we say if they depend continuously? That is to say, can matrices with similar entries have arbitrarily different minimum polynomials?

$$\begin{pmatrix} \mu & a \\ 0 & \mu \end{pmatrix}$$ has minimal polynomial $(t-\mu)^2$ unless $a=0$, in which case the minimal polynomial is $t-\mu$, so an arbitrarily small change in $a$ can change the minimal polynomial's degree, which we probably don't want to call a small change.

• Thanks! Perfect response – Adam Higgins Jul 9 '18 at 14:42

$\begin{pmatrix} 0 & \frac{1}{n} \\ 0 & 0 \end{pmatrix}$ has minimal polynomial $X^2$, but it converges to $0$, which has $X$ as its minimal polynomial.