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(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.

Jake Grimes
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  • To add onto this, I have seen several sources, such as the top answer to this question say that the right singular vectors are eigenvectors of the covariance matrix $X^TX$. But $X^TX$ can't be the covariance matrix because it's not normalized to have mean 0. What am I missing? – Jake Grimes Jan 11 '17 at 00:33

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