Unformatted text preview: m . n − 1 X k =1 k X i =1 2 m ≈ mn 2 CHALLENGE 5.11. • Suppose z = ± b − A ˜ x ± ≤ ± b − Ax ± for all values of x . Then by multiplying this inequality by itself we see that z 2 = ± b − A ˜ x ± 2 ≤ ± b − Ax ± 2 , so ˜ x is also a minimizer of the square of the norm. • Since QQ ∗ = I , we see that  Q ∗ y  2 2 = ( Q ∗ y ) ∗ ( Q ∗ y ) = y ∗ QQ ∗ y = y ∗ y =  y  2 2 , Since norms are nonnegative quantities, take the square root and conclude that  Q ∗ y  2 =  y  2 . • Suppose y 1 contains the Frst p components of the mvector y . Then ± y ± 2 2 = m X j =1  y j  2 = p X j =1  y j  2 + m X j = p +1  y j  2 = ± y 1 ± 2 2 + ± y 2 ± 2 2 ....
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 Fall '11
 Dr.Robin
 Multiplication, Complex number, Hilbert space, Matrix Factorizations, Q∗ y2

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