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What is the minimal polynomial for the zero matrix ? It cannot be $P_0(X)=0$ where $X$ is a matrix because this polynomial does not divide anything. Any polynomial of degree $0$ does not give $P_0(0)=0$ so is it a polynomial of degree $1$ ?

Are all minimal polynomials of degree $\geq 1$ ?

Kilkik
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    In my class we defined the minimal polynomial as the only monic polynomial of minimum degree such that $P(A) = 0$ (it turns out the can only be one). With this in mind the minimal polynomial of the zero matrix is $P(x) = x$. – Asinomás May 24 '21 at 20:12
  • That's what I thought, thanks ! – Kilkik May 24 '21 at 20:14
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    Note that every irreducible factor of the characteristic polynomial divides the minimal polynomial. Since the characteristic polynomial of the zero matrix is $(-1)^nx^n$, then the minimal polynomial must be of the form $x^k$ for some $k\geq 1$. The one of least degree that works is therefore $x$. – Arturo Magidin May 24 '21 at 20:16
  • Just a note: see here https://math.stackexchange.com/questions/495378/the-degree-of-zero-polynomial and https://math.stackexchange.com/questions/1796312/what-is-the-degree-of-the-zero-polynomial-and-why-is-it-so for a discussion of the degree of the zero polynomial - it depends on context whether you can/should call the zero polynomial a "polynomial of degree zero" – Mark Bennet May 24 '21 at 20:21

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