Editing Spectrum of a matrix

{{Short description|The set of the matrix's eigenvalues}}
In [[mathematics]], the '''spectrum''' of a [[matrix (mathematics)|matrix]] is the [[set (mathematics)|set]] of its [[eigenvalue]]s.<ref>{{harvtxt|Golub|Van Loan|1996|p=310}}</ref><ref>{{harvtxt|Kreyszig|1972|p=273}}</ref><ref>{{harvtxt|Nering|1970|p=270}}</ref>  More generally, if <math>T\colon V \to V</math> is a [[linear operator]] on any [[dimension (vector space)|finite-dimensional]] [[vector space]], its spectrum is the set of scalars <math>\lambda</math> such that <math>T-\lambda I</math> is not [[invertible function|invertible]]. The [[determinant]] of the matrix equals the product of its eigenvalues. Similarly, the [[Trace (linear algebra)|trace]] of the matrix equals the sum of its eigenvalues.<ref>{{harvtxt|Golub|Van Loan|1996|p=310|}}</ref><ref>{{harvtxt|Herstein|1964|pp=271–272}}</ref><ref>{{harvtxt|Nering|1970|pp=115–116}}</ref>
From this point of view, we can define the [[pseudo-determinant]] for a [[singular matrix]] to be the product of its nonzero eigenvalues (the density of [[multivariate normal distribution]] will need this quantity).

In many applications, such as [[PageRank]], one is interested in the dominant eigenvalue, i.e. that which is largest in [[absolute value]]. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.

== 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.

== Related notions ==
The [[eigendecomposition]] (or spectral decomposition) of a [[diagonalizable matrix]] is a [[matrix decomposition|decomposition]] of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

The [[spectral radius]] of a [[square matrix]] is the largest absolute value of its eigenvalues. In [[spectral theory]], the spectral radius of a [[bounded linear operator]] is the [[supremum]] of the absolute values of the elements in the spectrum of that operator.

== Notes ==
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== References ==
* {{ citation | first1 = Gene H. | last1 = Golub | first2 = Charles F. | last2 = Van Loan | year = 1996 | isbn = 0-8018-5414-8 | title = Matrix Computations | edition = 3rd | publisher = [[Johns Hopkins University Press]] | location = Baltimore }}
* {{ citation | first1 = I. N. | last1 = Herstein | year = 1964 | isbn = 978-1114541016 | title = Topics In Algebra | publisher = [[Blaisdell Publishing Company]] | location = Waltham }}
* {{citation | first1 = Erwin | last1 = Kreyszig | year = 1972 | isbn = 0-471-50728-8 | title = Advanced Engineering Mathematics | edition = 3rd | publisher = [[John Wiley & Sons|Wiley]] | location = New York | url-access = registration | url = https://archive.org/details/advancedengineer00krey }}
* {{ citation | first1 = Evar D. | last1 = Nering | year = 1970 | title = Linear Algebra and Matrix Theory | edition = 2nd | publisher = [[John Wiley & Sons|Wiley]] | location = New York | lccn = 76091646 }}

[[Category:Matrix theory]]


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