Residual growth in an indefinite steepest-descent iteration ($A=\left(\begin{array}{cc}2& 3\\ -5& -2\end{array}\right)$)

For this nonsymmetric matrix, consider the formal residual update $r^{+}=r-(r^{T}r)/(r^{T}Ar)Ar$, with nonzero $r=(x,y{)}^T$. Put $D=2x^2-2xy-2y^2$ and $B=5x^2+4xy+3y^2$. Algebra gives $r^{+}=(B/D)(-y,x{)}^T$ when $D\ne 0$. But $B+D=7(x+y/7{)}^2+6y^2/7>0$ and $B-D=3(x+y{)}^2+2y^2>0$. Hence $B>|D|$ and $\Vert r^{+}\Vert >\Vert r\Vert$ for every defined nonzero step. At $D=0$, the quotient is undefined. Thus the iteration cannot converge to zero linear-system residual from any nonzero initial residual. This concrete divergence proof distinguishes failure of an algorithm from mere failure to meet a sufficient theorem's hypotheses.