Unshifted QR iteration, shifted QR iteration, deflation, and the Wilkinson shift

Let $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and let $A_0\in M_n(\mathbb{F})$.

1. The unshifted QR iteration factors $A_k=Q_k{R}_k$ with $Q_k$ unitary and $R_k$ upper triangular, then sets $A_{k+1}:=R_k{Q}_k\mathrm{.}$
2. Given shifts ${\mu }_k\in \mathbb{F}$, the shifted QR iteration factors $A_k-{\mu }_{k}I=Q_k{R}_k$ and sets $A_{k+1}:=R_k{Q}_k+{\mu }_{k}I\mathrm{.}$
3. Given a declared tolerance ${\tau }_k\ge 0$, numerical deflation at index $j$ occurs when $\mathrm{\mid }(A_k{)}_{j+1,j}\mathrm{\mid }\le {\tau }_k$: that entry is then replaced by $0$, splitting the matrix into the two diagonal blocks separated at index $j$. Exact deflation is the special case ${\tau }_k=0$.
4. For a real symmetric tridiagonal matrix, the Wilkinson shift is the eigenvalue of the trailing $2\times 2$ principal block that is nearer to the bottom-right entry; away from a tie, this eigenvalue is unique.