MATH235 Study Guide - Final Guide: Orthogonal Matrix, Diagonal Matrix, Symmetric Matrix

• School: University of Waterloo
• Department: Mathematics
• Course: MATH235
• Professor: Ghazal Geshnizjani

Document Summary

Assignment 7 solutions: for each of the following symmetric matrices, nd an orthogonal matrix p and diagonal matrix d such that p t ap = d. Hence, the eigenvalues are 1 = 4 and 2 = 9. Solution: we have c( ) = det(a i) = (cid:12)(cid:12)(cid:12)(cid:12) For 1 = 4 we get a 1i = (cid:20)2 3. For 2 = 9 we get a 2i = (cid:20) 3. 1/ 2(cid:21) is orthogonal and p t ap = (cid:20)4 0. = 2 13 + 36 = ( 4)( 9) After normalizing, the basis vectors for the eigenspaces form an orthonormal basis for r2. (b) a = . Hence, the eigenvalues are 1 = 0, 2 = 10, and 3 = 1. Thus, a basis for e 1 is . For 1 = 0 we get a 1i = . For 2 = 10 we get a 2i = . For 3 = 1 we get a 3i = .