The problem formulation is to show that PCA involves choosing a vector u, so as to minimize the sum of the squares of the projection errors(of the training examples x onto u),subject to u'*u=1 x(i) = R mx1 , u is a vector I.e minimize sum over i, Norm square of(projection of x(i) - x(i)) subject to u'*u=1
How do i represent the projection of x(i) onto the vector u and formulate the lagrangian?
If $u$ is a vector of norm 1 (i.e. $u'u = 1$) then the projection of $x(i)$ on $u$ is the vector $\langle x(i),u\rangle u$ with $\langle x(i),u\rangle$ the scalar product of the two vectors.
The PCA consists in minimizing the error, as you said, $$\sum_i \|x(i)-\langle x(i),u\rangle u\|^2, $$ or equivalently $$\text{maximize }\sum_i \langle x(i), u\rangle^2$$ This can be solved, not using the Lagrangian but the following matrix formulation. If $X$ is the matrix of row $x(i)$, then the problem is to $$\text{maximize }\| Xu\|^2$$ which is a simple eigenvalue problem...