Editing Spectrum of a matrix (section)

{{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.