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Suppose you're given a minimal polynomial, is there an efficient and systematic (!) way to find a matrix with given minimal polynomial?

For example for $n \in \mathbb{N}$ find an $n \times n$-Matrix such that its minimal polynomial is $(X - 1)^k$ for a given $1 \leq k \leq n$.

Say we even limit ourselves to $n=5$ and $k=4$, is there a better way to find such a Matrix without using just trial and error?

Dedekind
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  • I just recently learned about the companion matrix but would it also work if the minimal polynomial isn't of degree n? – Dedekind Mar 04 '22 at 17:17
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    Google "companion matrix" – DonAntonio Mar 04 '22 at 17:21
  • I think I know how a companion matrix works but how would a $5 \times 5$ companion matrix look for the polynomial $p(X) := (X-1)^4 = X^4 - 4X^3 + 6X^2 - 4X + 1$? In our lecture and on Wikipedia it's only defined if we have a polynomial of the form $X^5 + a_4 X^4 + ... + a_0$ which $p(X)$ doesn't fulfill. What am I missing? – Dedekind Mar 04 '22 at 17:27
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    If the polynomial is factored, use Jordan form. For $(X-1)^4$ in a $5x5$, put five $1$'s on the main diagonal, and then three $1$'s just above those in locations (2,1), (3,2) and (4,3), with $0$'s everywhere else. – Ned Mar 04 '22 at 21:02

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If you have a matrix $M$ such that $(M-I)^k=0$, then $$ M=I+N $$ where $N$ satisfies $N^k=0$. One such $N$ is a $k\times k$ block matrix $$ \left[\begin{array}{k,k}0 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & 1 \\ 0 & 0 & 0 & 0 & \cdots & 0\end{array}\right] $$ You can generalize this idea through Jordan Canonical Form.

Disintegrating By Parts
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