Editing Spectrum of a matrix (section)

== Definition ==

Let ''V'' be a finite-dimensional [[vector space]] over some [[field (mathematics)|field]] ''K'' and suppose ''T'' : ''V'' → ''V'' is a linear map. The ''spectrum'' of ''T'', denoted σ<sub>''T''</sub>, is the [[multiset]] of [[root of a polynomial|roots]] of the [[characteristic polynomial]] of ''T''. Thus the elements of the spectrum are precisely the eigenvalues of ''T'', and the multiplicity of an eigenvalue  ''λ'' in the spectrum equals the dimension of the [[generalized eigenspace]] of ''T'' for ''λ'' (also called the [[algebraic multiplicity]] of ''λ'').

Now, fix a [[basis (linear algebra)|basis]] ''B'' of ''V'' over ''K'' and suppose ''M'' ∈ Mat<sub>''K''{{hairsp}}</sub>(''V'') is a matrix. Define the linear map ''T'' : ''V'' → ''V'' pointwise by ''Tx'' = ''Mx'', where on the right-hand side ''x'' is interpreted as a column vector and ''M'' acts on ''x'' by [[matrix multiplication]]. We now say that ''x'' ∈ ''V'' is an [[eigenvector]] of ''M'' if ''x'' is an eigenvector of ''T''. Similarly, λ ∈ ''K'' is an eigenvalue of ''M'' if it is an eigenvalue of ''T'', and with the same multiplicity, and the spectrum of ''M'', written σ<sub>''M''</sub>, is the multiset of all such eigenvalues.