If ${\chi }_T(x)={\prod }_{i<n}(x-{\lambda }_i)$ in $F[x]$, then ${\chi }_{p(T)}(y)={\prod }_{i<n}(y-p({\lambda }_i))$ for every $p\in F[x]$: the eigenvalues of $p(T)$ are $p({\lambda }_i)$, counted with algebraic multiplicity

Given: $T$, the displayed split factorization of ${\chi }_T$, and $p\in F[x]$.

Polynomial evaluation is $p(T)={\sum }_k{a}_k{T}^k$ (Polynomial evaluation at an endomorphism: $p(T)={\sum }_k{a}_k{T}^k$). Matrix representation sends sums and scalar multiples to matrix sums and scalar multiples ($T\mapsto [T{]}_{\mathcal{B}}^{\mathcal{C}}$ is a vector-space isomorphism $\mathcal{L}(V,W)\cong M_{m\times n}(F)$) and composites to matrix products ($[S\circ T{]}_{\mathcal{B}}^{\mathcal{D}}=[S{]}_{\mathcal{C}}^{\mathcal{D}}[T{]}_{\mathcal{B}}^{\mathcal{C}}$).

The characteristic polynomial of an operator is computed in any basis and is $1$ in dimension zero (The basis-independent characteristic polynomial ${\chi }_T$ of an endomorphism of a finite-dimensional space, including ${\chi }_T=1$ in dimension zero).

Roots of ${\chi }_T$ are exactly eigenvalues (For every finite-dimensional space, ${\sigma }_F(T)$ is exactly the set of roots in $F$ of ${\chi }_T$).

An independent subset of a finite-dimensional space extends without Choice to a basis (If ${\dim }_{F}V=n$ and $U$ is a linear subspace of $V$, then $U$ is finite-dimensional, ${\dim }_{F}U\le n$, and ${\dim }_{F}U=n$ if and only if $U=V$, clause 3), whose representing matrix records the coordinate columns of the images (Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases).

A block-triangular characteristic polynomial is the product of the characteristic polynomials of its diagonal blocks (The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks).

The ring $F[x]$ is a unique factorisation domain, hence an integral domain and admits cancellation of nonzero polynomials (For every field $F$, $F[x]$ is a unique factorisation domain).

Algebraic multiplicity is the exponent of a root's linear factor in the characteristic polynomial (Algebraic multiplicity as the exponent of $x-\lambda$ in ${\chi }_T$, and geometric multiplicity as $\dim E_{\lambda }(T)$).