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As explained in amoeba's beautiful answer here one can use a singular value decomposition of the data matrix, $\mathbf{X} = \mathbf{USV}^\top$, to do a principal component analysis, if it is assumed that the data matrix $\mathbf{X}$ is centered. The principal directions will then be given as the columns of $\mathbf{V}$.

Now, I have come across some statistical analysis where the data matrix was centered and standardized for the data to have unit variance before being used in a singular value decomposition. The analysis then proceeded in using the columns of $\mathbf{V}$ as principal directions.

Can any of you think of a reason to do this standardization step before running the singular value decomposition? Is the result of such a procedure in terms of what PCA should tell you even meaningful?

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  • $\begingroup$ This is indeed a duplicate but +1 for a clear and well-formulated question. $\endgroup$
    – amoeba
    Sep 15, 2016 at 17:02

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In the context of PCA, the reason for making all the variables have the same variance (by standardizing them) is simple: to make them all be treated as equally important. PCA will treat variables with greater variance as more important, since it tries to account for the greatest possible total variance with each successive component.

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