We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write $$ J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2. $$
We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by $$ \mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2, $$ since the expectation value of $$ \mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0, $$ where we have used the fact that our data are centered.
Recalling our definition of the covariance as $$ \boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T], $$ we have thus that $$ \mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0. $$
We are almost there, we have obtained a relation between minimizing the reconstruction error and the variance and the covariance matrix. Minimizing the error is equivalent to maximizing the variance of the projected data.