Spectrum of a matrix: Difference between revisions

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In mathematics, the spectrum of a (finite) matrix is the multiset of its eigenvalues. This notion can be extended to the spectrum of an operator in the infinite-dimensional case.

The determinant equals the product of the eigenvalues. Similarly, the trace equals the sum of the eigenvalues. From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of all the nonzero eigenvalues (the density of multivariate normal distribution will need this quantity).

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_T, 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_K(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_M, is the multiset of all such eigenvalues.

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-In [[mathematics]], the '''spectrum''' of a (finite-dimensional) matrix is the [[set (mathematics)|set]] of its [[eigenvalue]]s.  This notion can be extended to the [[spectrum of an operator]] in the infinite-dimensional case.
+In [[mathematics]], the '''spectrum''' of a (finite) matrix is the [[multiset]] of its [[eigenvalue]]s.  This notion can be extended to the [[spectrum of an operator]] in the infinite-dimensional case.
 The [[determinant]] equals the product of the eigenvalues. Similarly, the [[Trace (linear algebra)|trace]] equals the sum of the eigenvalues.
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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. An ''eigenvector'' of ''T'' is a non-zero vector ''x'' ∈ ''V'' such that ''Tx''{{=}}λ''x'' for some λ∈''K''. The value λ is called an ''eigenvalue'' of ''T'' and the set of all such eigenvalues is called the ''spectrum'' of ''T'', denoted σ<sub>''T''</sub>.
+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 the spectrum of ''M'', written σ<sub>''M''</sub>, is the set of all such eigenvalues.
+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.
 [[Category:Matrix theory]]