The basic idea of the QR algorithm is to decompose the matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R, and then to multiply the factors in reverse order to obtain a new matrix A' = RQ. The process is repeated until convergence.
Here's a write-up based on the book:
Parlett, B. N. (1998). The symmetric eigenvalue problem. SIAM. parlett the symmetric eigenvalue problem pdf
A very specific request!