Mastering SciPy

Matrix factorizations based on eigenvalues

In this category, we have two kinds of factorizations on square matrices: Spectral and Schur decompositions (although, technically, a spectral decomposition is a special case of Schur decomposition). The objective of both is initially to present the eigenvalues of one or several matrices simultaneously, although they have quite different applications.

Spectral decomposition

We consider the following four cases:

• Given a square matrix A, we seek all vectors v (right eigenvectors) that satisfy A ● v = m ● v for some real or complex value m (the corresponding eigenvalues). If all eigenvectors are different, we collect them as the columns of matrix V (that happens to be invertible). Their corresponding eigenvalues ...