Diagonalization Strategy
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Diagonalization algorithms usually have three significant steps. The
first step is to convert the matrix to upper Hessenberg form, where all
matrix elements below the subdiagonal are zeros. A sequence of
similarity transforms can put a matrix into Hessenberg form. The second
step is the Schur decomposition, where an algorithm uses a sequence of
similarity transforms to convert the matrix to upper triangular form per
the Triangularization Theorem. The Schur decomposition requires an
iterative algorithm. The final step is to find the eigenvectors from the
Schur vectors and the eigenvectors of the upper triangular matrix. The
triangular structure of the matrix allows for computing the eigenvectors
with a direct calculation.