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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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