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11. (Strang 4.2.21, 22, 28) Consider a matrix A with linearly independent columns and its projection matrix, P = A(AT A)-'A". (a) Prove that P² = P, by multiplying P by itself and simplifying. (b) Prove that P is symmetric, by computing P"; recall that the inverse of a symmetric matrix is also symmetric. (c) Use the results of (a) and (b) to prove that the diagonal entry P22 must always be equal to the square of the length of the second column of P.

11. (Strang 4.2.21, 22, 28) Consider a matrix A with linearly independent columns and its projection matrix, P = A(AT A)-'A". (a) Prove that P² = P, by multiplying P by itself and simplifying. (b) Prove that P is symmetric, by computing P"; recall that the inverse of a symmetric matrix is also symmetric. (c) Use the results of (a) and (b) to prove that the diagonal entry P22 must always be equal to the square of the length of the second column of P.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.3: The Gram-schmidt Process And The Qr Factorization
Problem 23EQ
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11. (Strang 4.2.21, 22, 28) Consider a matrix A with linearly independent columns and its projection matrix, P = A(AT A)-'A". (a) Prove that P² = P, by multiplying P by itself and simplifying. (b) Prove that P is symmetric, by computing P"; recall that the inverse of a symmetric matrix is also symmetric. (c) Use the results of (a) and (b) to prove that the diagonal entry P32 must always be equal to the square of the length of the second column of P.
Transcribed Image Text:11. (Strang 4.2.21, 22, 28) Consider a matrix A with linearly independent columns and its projection matrix, P = A(AT A)-'A". (a) Prove that P² = P, by multiplying P by itself and simplifying. (b) Prove that P is symmetric, by computing P"; recall that the inverse of a symmetric matrix is also symmetric. (c) Use the results of (a) and (b) to prove that the diagonal entry P32 must always be equal to the square of the length of the second column of P.
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