Exercise 22.11. This exercise leads through a direct argument, without reference to Gram-Schmidt, in the case...

 

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Exercise 22.11. This exercise leads through a direct argument, without reference to Gram-Schmidt, in the case of invertible square matrices that QR-decompositions are uniquely determined. That is, if an invertible n x n matrix A is equal to QR and to Q'R' for orthogonal Q and Q' and upper triangular R and R' with positive diagonal entries then Q = Q' and R = R'. -1 (a) Show Q-¹Q' = RR'¯¹, and that the left side is orthogonal and the right side is upper triangular with QR, multiply both sides by Q¹ on positive entries. (Hint for the equality: beginning with Q'R' the left and R¹ on the right.) = (b) Show that if Q−¹Q' = In and RR'¯¹ In then Q Q' and R = R'. (c) Show that if an orthogonal n × n matrix M is also upper triangular with positive diagonal entries then it equals In, and combine this with (a) and (b) to conclude that Q = Q' and R = R'. (Hint: first show the left column of M must be e₁ [and not -e₁, for instance] and then use that knowledge to deduce that the second column must be e2, and so on. Think about n = 2 before thinking about general n.) = - Exercise 22.11. This exercise leads through a direct argument, without reference to Gram-Schmidt, in the case of invertible square matrices that QR-decompositions are uniquely determined. That is, if an invertible n x n matrix A is equal to QR and to Q'R' for orthogonal Q and Q' and upper triangular R and R' with positive diagonal entries then Q = Q' and R = R'. -1 (a) Show Q-¹Q' = RR'¯¹, and that the left side is orthogonal and the right side is upper triangular with QR, multiply both sides by Q¹ on positive entries. (Hint for the equality: beginning with Q'R' the left and R¹ on the right.) = (b) Show that if Q−¹Q' = In and RR'¯¹ In then Q Q' and R = R'. (c) Show that if an orthogonal n × n matrix M is also upper triangular with positive diagonal entries then it equals In, and combine this with (a) and (b) to conclude that Q = Q' and R = R'. (Hint: first show the left column of M must be e₁ [and not -e₁, for instance] and then use that knowledge to deduce that the second column must be e2, and so on. Think about n = 2 before thinking about general n.) = -

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A We are given that A QR and A QR where Q and Q are orthogonal matrices ie QQ I and R and R are uppe... View the full answer