Principal Component Analysis with Missing Values

Let us consider the same problem when the data matrix has missing entries. In the following there are $N=9$ observed values and $6$ missing values marked with a question mark (?):

$$\mathbf{X}= \begin{bmatrix} -1 & +1 & 0 & 0 & ? \\ -1 & +1 & ? & ? & 0 \\ ? & ? & -1 & +1 & ? \end{bmatrix}\, .$$ (8)

We would like to find $\mathbf{A}$ and $\mathbf{S}$ such that $\mathbf{X}\approx \mathbf{A}\mathbf{S}$ for the observed data values. The rest of the product $\mathbf{A}\mathbf{S}$ represents the reconstruction of missing values.