Exercise 8.C.4

Proof.

Let

Then

Define $T\in L({ℂ}^4)$ by

Then $M(T)=A$ and the eigenvalues of $T$ are thus $1$ and $5$ (the entries on the diagonal). Now 8.36 implies that the minimal polynomial of $T$ is a polynomial multiple of $(z-5)(z-1)$. The previous work shows that $(T-5I)(T-I)\ne 0$ and ${(T-5I)}^2(T-I)=0$. Hence ${(z-5)}^2(z-1)$ is the minimal polynomial of $T$. By Exercise 11 in section 8B, the multiplicity of $1$ is $1$ and of $5$ is $3$. Thereby the characteristic polynomial of $T$ is $(z-1){(z-5)}^3$. □