Editing Companion matrix (section)

== Multiplication map on a simple field extension ==
Consider a polynomial <math>p(x)=x^n + c_{n-1}x^{n-1}  + \cdots +  c_1 x + c_0 </math> with coefficients in a [[Field (mathematics)|field]] <math>F</math>, and suppose <math>p(x)</math> is [[Irreducible polynomial|irreducible]] in the [[polynomial ring]] <math>F[x]</math>. Then [[Simple extension|adjoining a root]] <math>\lambda</math> of <math>p(x)</math> produces a [[field extension]] <math>K=F(\lambda) \cong F[x]/(p(x))</math>, which is also a vector space over <math>F</math> with standard basis <math>\{1,\lambda,\lambda^2,\ldots,\lambda^{n-1}\} </math>. Then the <math>F</math>-linear multiplication mapping
{{ block indent | em = 1.5 | text = <math>m_{\lambda}:K\to K</math>  defined by  <math>m_\lambda(\alpha) = \lambda\alpha</math> }}
has an ''n'' × ''n'' matrix <math>[m_\lambda]</math> with respect to the standard basis. Since <math>m_\lambda(\lambda^i) = \lambda^{i+1}</math> and <math>m_\lambda(\lambda^{n-1}) = \lambda^n = -c_0-\cdots-c_{n-1}\lambda^{n-1}</math>, this is the companion matrix of <math>p(x)</math>:
<math display="block">[m_\lambda] = C(p).</math>
Assuming this extension is [[Separable extension|separable]] (for example if <math>F</math> has [[characteristic zero]] or is a [[finite field]]), <math>p(x)</math> has distinct roots <math>\lambda_1,\ldots,\lambda_n </math> with <math>\lambda_1  = \lambda</math>, so that
<math display="block">p(x)=(x-\lambda_1)\cdots (x-\lambda_n),</math>
and it has [[splitting field]] <math>L = F(\lambda_1,\ldots,\lambda_n)</math>. Now <math>m_\lambda</math> is not diagonalizable over <math>F</math>; rather, we must [[Extension of scalars|extend]] it to an <math>L</math>-linear map on <math>L^n \cong L\otimes_F K</math>, a vector space over <math>L</math> with standard basis <math>\{1{\otimes} 1, \, 1{\otimes}\lambda, \, 1{\otimes}\lambda^2,\ldots,1{\otimes}\lambda^{n-1}\}  </math>, containing vectors <math>w = (\beta_1,\ldots,\beta_n) = \beta_1{\otimes} 1+\cdots+\beta_n{\otimes} \lambda^{n-1} </math>. The extended mapping is defined by <math>m_\lambda(\beta\otimes\alpha) = \beta\otimes(\lambda\alpha)</math>. 

The matrix <math>[m_\lambda] = C(p)</math> is unchanged, but as above, it can be diagonalized by matrices with entries in <math>L</math>: 
<math display="block">[m_\lambda]=C(p)= V^{-1}\! D V,</math>
for the diagonal matrix <math>D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)</math> and the [[Vandermonde matrix]] ''V'' corresponding to <math>\lambda_1,\ldots,\lambda_n\in L </math>. The explicit formula for the eigenvectors (the scaled column vectors of the [[Vandermonde matrix#Inverse Vandermonde matrix|inverse Vandermonde matrix]] <math>V^{-1}</math>) can be written as:
<math display="block">\tilde w_i = 
\beta_{0i} {\otimes} 1+\beta_{1i} {\otimes} \lambda  +\cdots + \beta_{(n-1)i} {\otimes} \lambda^{n-1}
= \prod_{j\neq i} (1{\otimes}\lambda - \lambda_j{\otimes} 1) </math>
where <math>\beta_{ij}\in L </math> are the coefficients of the scaled Lagrange polynomial
<math display="block">\frac{p(x)}{x-\lambda_i}
= \prod_{j\neq i} (x - \lambda_j) = \beta_{0i} + \beta_{1i} x + \cdots + \beta_{(n-1)i} x^{n-1}.</math>