5. Triangularisation with an orthogonal matrix Example '19 in the Study Guide {pages 21-23 of Topic ?} shows the triangularisation procedure for a matrix. Consider the following matrix A, which also has eigenvalues 1, 1 and 5. 3 2 2 A = 1 2 1 1 1 2 i. Construct a matrix S such that it is an orthogonal matrix with the rst column corresponding 1 with the eigenvector {I . Calculate 31113. 1 le [1.41 1 and 5 by determining and then simplifying its characteristic equation. ii. Show that for the resulting matrixI S 4113 = [ :|, the 2 x 2 matrix A1 has eigenvalues iii. Find the eigenvector from A1 corresponding with A = 1 and then construct an orthogonal 2 X 2 matrix Q where the rst column is based on your eigenvector. Hence construct the matrix 100 R= HQ iv. Calculate P = SR. Show that P is an orthogonal matrix and verify that P'1AP is upper triangular. 5 marks