Spectrum of a matrix: Difference between revisions
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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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In [[mathematics]], the '''spectrum''' of a (finite |
In [[mathematics]], the '''spectrum''' of a (finite) matrix is the [[]] of its [[eigenvalue]]s. This notion can be extended to the [[spectrum of an operator]] in the infinite-dimensional case. |
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The [[determinant]] equals the product of the eigenvalues. Similarly, the [[Trace (linear algebra)|trace]] equals the sum of the eigenvalues. |
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 == |
== Definition == |
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Let ''V'' be a finite-dimensional [[vector space]] over some field ''K'' and suppose ''T'': ''V'' → ''V'' is a linear map. |
Let ''V'' be a finite-dimensional [[vector space]] over some field ''K'' and suppose ''T'': ''V'' → ''V'' is a linear map. '''' of ''T'' '''' ''''. of ''T'' and the of the '''' '''' ''''. |
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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 |
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 of all such eigenvalues. |
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[[Category:Matrix theory]] |
[[Category:Matrix theory]] |
Revision as of 14:57, 5 September 2013
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 σT, 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∈MatK(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 σM, is the multiset of all such eigenvalues.