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# 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 ...

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