In Exercise 21.11, you may use the MATLAB routine from the Google PageRank book to verify the results.
In the proof of Theorem 4.7.1 on page 46 of the Google PageRank book, verify the computation that Q^(-1)SQ = the block matrix consisting of y^T e in the upper left corner, y^T SX in the upper right corner, Y^T e in the lower left corner, and Y^T SX in the lower right corner.