  We propose a new way of visualising the dynamics of iterative eigenvalue
algorithms such as the QR algorithm, over the important special case of PSD
(positive semi-definite) matrices. Many subtle and important properties of such
algorithms are easily found this way. We believe that this may have pedagogical
value to both students and researchers of numerical linear algebra. The fixed
points of iterative algorithms are obtained visually, and their stability is
analysed intuitively. It becomes clear that what it means for an iterative
eigenvalue algorithm to "converge quickly" is an ambiguous question, depending
on whether eigenvalues or eigenvectors are being sought. The presentation is
likely a novel one, and using it, a theorem about the dynamics of general
iterative eigenvalue algorithms is proved. There is an accompanying video
series, currently hosted on Youtube, that has certain advantages in terms of
fully exploiting the interactivity of the visualisation.
