A visualisation of the dynamics of the unshifted QR algorithm for PSD matrices is given here, with a remark as to how shifts help: https://www.youtube.com/watch?v=XGAaHoppcvE
The video is by me.
The argument is given for PSD matrices. For the set of PSD matrices similar to some PSD matrix $M$ (where $M$ is not a multiple of the identity matrix), there are two fixed points of the unshifted QR algorithm. One of these fixed points is stable, while the other is unstable. The introduction of shifts eliminates the unstable fixed point, leaving behind only stable fixed points. Unstable fixed points are undesirable because they compromise the speed of the algorithm. The shifts must depend discontinuously on the matrix; however this does not alter the practicality of the algorithm.
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