Annihilating polynomial and minimal polynomial

[finite iris]

Before this, @river quail suggested something which I'm doing right now, but slightly differently:

[rustic obsidian]

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[finite iris]

(A * v * B doesn't make sense anyways)

[finite iris]

(where e_i is the unit vector with 1 at position i)

[river quail]

If P and Q are polynomials with coefficients in $K$ then $P(A) Q(A)=Q(A) P(A)$ for a matrice $A$

[hot sand]

Rotor 😑

[river quail]

Polynomials in a matrice commute

And if A and B commute then their kernels are stable from one matrice to another ie if Av=0 then ABv=0

And in my opinion you are over complicating, what you are doing is correct, from what I understand you want to prove that if $(e_i){1 \le i \le n}$ is the canonical basis of $K^{n}$ then $(\mu{A,v_1} … \mu_{A, v_n})(A) . e_i=0$ for all $i \in {1,…,n}$ from my understanding

[hot sand]

Rotor 😑

[river quail]

But doing so for any basis works so why not take the given basis ?

[finite iris]

I picked e_i because A * e_i represents the i'th column of our matrix that needs to be 0

Picking v instead of e_i doesn't do this

Ah

+close