testing testing

 

Now expand:

∣xn−x~n∣2=(xn−u1(u1Txn))T(xn−u1(u1Txn))

 

$|x_n - \tilde{x}_n|^2 = \left( x_n - u_1 (u_1^T x_n) \right)^T \left( x_n - u_1 (u_1^T x_n) \right)$

 

Expanding the terms:

=xnTxn−2xnTu1(u1Txn)+(u1(u1Txn))T(u1(u1Txn))

 

$= x_n^T x_n - 2 x_n^T u_1 (u_1^T x_n) + \left( u_1 (u_1^T x_n) \right)^T \left( u_1 (u_1^T x_n) \right) $

 

For the last term, note that:

 

(u1(u1Txn))T(u1(u1Txn))=(u1Txn)2(u1Tu1)

 

$\left( u_1 (u_1^T x_n) \right)^T \left( u_1 (u_1^T x_n) \right) = (u_1^T x_n)^2 (u_1^T u_1)$