Spectrum of a matrix

In mathematics, the spectrum of a matrix is the set of its eigenvalues.^[1][2][3] More generally, if ${\displaystyle T\colon V\to V}$ is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars ${\displaystyle \lambda }$ such that ${\displaystyle T-\lambda I}$ is not invertible. The determinant of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues.^[4][5][6] 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.