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I am working on a dataset with many categorical variables for a clustering problem. I've done one-hot encoding where a categorical column with 5 levels will become 5 columns, each has the standard deviation of 1 after standardization. I am thinking of using PCA to cluster data to describe characteristics of data in each cluster.

My question is: can I apply PCA directly to the one-hot encoded dataset? My concern is that PCA assumes equal weight among all features and the original categorical variable (now in 5 columns) will have 5 times the weight than a normal numerical variable. Is it a problem? One thing to do, I think, is to divide the values in the 5 columns by the square root of 5, then the sum of 5 columns' variance is back to 1. Does this make sense?

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