Properties of the polynomials of matrices

Question

For a complex square matrix $M$, let us say $n\times n$, we can define the $\mathbb{C}$-algebra homomorphism,

$$ \phi_M:\mathbb{C}[x]\rightarrow Y_M:=\{N\in\mathbb{M}_n(\mathbb{C})|[M,N]=0\}$$ $$p(x)\mapsto p(M)$$

I would like to know when it is surjective (onto $Y_M$), I already know that for a diagonalizable matrix $M$ it is the case iff $M$ has distinct eigenvalues, but for the case where $M$ has a nilpotent part I do not know any conditions.

$P(M)$ always commute with $M$. I did not understand what you are looking for — Lelouch, Oct 21 '22 at 15:40

Ben Grossmann · Accepted Answer · 2022-10-21T16:15:57.613

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As can be shown using the Cayley Hamilton theorem or other means, the image of this homomorphism is a subspace of $\Bbb C^{n \times n}$ with dimension at most equal to $n$. So to answer your question, this homomorphism is never surjective except in the trivial cases of $n=0,1$.

For a given $M \in \Bbb C^{n \times n}$, the image of this homomorphism attains its largest possible value of $n$ if and only if $M$ is a non-derogatory matrix. As is noted in this linked answer, this is also equivalent to the image of the homomorphism being the entire set of matrices that commute with $M$.

edited Oct 21 '22 at 16:15

answered Oct 21 '22 at 15:59

Ben Grossmann

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I guess he means surjective in the subspace of matrices that commute with $M$ and not in $\Bbb C^{n\times n}$ – InsideOut Oct 21 '22 at 16:05
@InsideOut That's an interesting guess, I hope we get some clarification – Ben Grossmann Oct 21 '22 at 16:14
PS: my mistake I wrote “he” in place of OP. I’m sorry! – InsideOut Oct 21 '22 at 16:17
Yes, that's what I meant, surgective onto the matrices that commute with $M$. – Arararararagi-kun Oct 21 '22 at 16:55
@A Thanks for clarifying. In any case, you have your answer: it is surjective if and only if $M$ is non-derogatory. – Ben Grossmann Oct 21 '22 at 17:04
The post really solves my problem, but it is not clear (for me) the last equivalence, could I get some reference please? – Arararararagi-kun Oct 22 '22 at 02:32
“Matrix Analysis” by Horn and Johnson 2nd edition has a proof of this in chapter 3 – Ben Grossmann Oct 22 '22 at 16:43

Properties of the polynomials of matrices

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