(The following is motivated by the singular value decomposition:)

Say we have an N x D matrix $A$. The first principal component $v_1$ is the unit-length vector of dimension D which maximizes the variances of the projections $v_1^Tx_i$. But I have also heard it described as the vector that maximizes the sum-of-squares projections: i.e. $v_1 = argmax_v|Av|$. Why are these two interpretions equivalent? I played with some numbers and found maximizing the variance is equivalent to maximizing $n\sum_{i}(v^Tx_i)^2 - (\sum_iv^Tx_i)^2$. But it isn't clear to me that the mean projection should be 0.