Spectrum of a matrix

In mathematics, the spectrum of a (finite-dimensional) matrix is the set 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. An eigenvector of T is a non-zero vector x ∈ V such that Tx=λx 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_T.

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 the spectrum of M, written σ_M, is the set of all such eigenvalues.