A deflation found in a Hessenberg (or tridiagonal in symmetric case) matrix.
Given a Hessenberg matrix,
\[
\begin{bmatrix}
H_{11} & H_{12} & H_{13}\\
0 & H_{22} & H_{23}\\
0 & 0 & H_{33}
\end{bmatrix}
\]
- \(u_l\) is the upper left hand corner index of H22;
- \(l_r\) is the lower right hand corner index of H22
Deflation of an upper Hessenberg matrix splits it into multiple smaller upper Hessenberg
matrices
when the sub-diagonal entries are sufficiently small. For example, suppose
\[
H = \begin{bmatrix}
1 & 2 & 3 & 4 & \\
5 & 6 & 7 & 8 & \\
0 & 9 & 10 & 11 & \\
0 & 0 & 12 & 13 &
\end{bmatrix}
\]
We can split
H into
H1 and
H2, so that
\[
H_1 = \begin{bmatrix}
1 & 2 \\
5 & 6 \\
\end{bmatrix}
\]
\[
H_2 = \begin{bmatrix}
10 & 11 \\
12 & 13 \\
\end{bmatrix}
\]