Deflation in the QR iteration

We need to introduce the concept of deflation before continuing.

The goal of the QR iteration is to produce a matrix that is upper triangular. This happens in stages. Let’s assume that we have applied the QR iteration to, approximately, transform matrix $A$ to a block upper triangular matrix of the form:

$$A=\left(\begin{array}{cc}{A}_{11}& A_{12}\\ 0& A_{22}\end{array}\right)$$

We can check that the eigenvalues of $A$ are equal to those of $A_{11}$ and $A_{22}$:

$$\lambda (A)=\lambda (A_{11})\cup \lambda (A_{22})$$

So if we have transformed $A$ to this form, we can get all its eigenvalues by running the QR iteration on the smaller matrices $A_{11}$ and $A_{22}$. This process of progressively reducing $A$ to smaller matrices is how we obtain all the eigenvalues of $A$.