david/Reformulation of the 2S conjecture in canonical Shu-Osher form

The form appearing in my thesis involves the conditions that \(I+rK\) be invertible and

\[ b^T e = 1 \] \[ b^T A e = \frac{1}{2} \] \[ K(I+rK)^{-1} \ge 0\] \[rK(I+rK)^{-1} \le 1\] where \[K = \left(\begin{array}{cc} A & 0 \\ b^T & 0 \end{array}\right).\]

Without loss of generality we can use the canonical Shu-Osher form with \(\alpha=r\beta\), where \[\beta = \left(\begin{array}{cc} \beta_A & 0 \\ \beta_b^T & 0 \end{array}\right).\]

Using the relations \[b^T = \beta_b^T(I-r\beta_A)^{-1}\] \[A = \beta_A(I-r\beta_A)^{-1}\]

we obtain the equivalent conditions that \(I-r\beta\) be invertible and \[ \beta_b^T(I-r\beta_A)^{-1}e = 1 \] \[ \beta_b^T(I-r\beta_A)^{-2}\beta_A e = \frac{1}{2}\] \[\beta\ge0\] \[r\beta e \le 1.\] In this form, the inequalities are simple but the equalities are more complex.