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spectrum has to be a multiset rather than a set if you want the stated relations with determinant and trace |
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== Definition ==
Let ''V'' be a finite-dimensional [[vector space]] over some field ''K'' and suppose ''T'': ''V'' → ''V'' is a linear map. The ''spectrum'' of ''T'', denoted σ<sub>''T''</sub>, is the multiset of 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 ''B'' of ''V'' over ''K'' and suppose ''M''∈Mat<sub>''K''</sub>(''V'') is a matrix. Define the linear map ''T'': ''V''→''V'' point-wise 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.
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