Do I use the mean vector from my training set to center my testing set when dimension reducing for classification?
Do I use the mean vector from my training set to center my testing set when dimension reducing for classification?: Yes. 

Test set must not be combined with training set in any step of calculating the reduced dimension space. Characteristics of final space is determined by training set and test set just follows that i.e. the mean-adjusting step uses training mean.

You just calculate the final eigenvectors matrix $E$ (whose dimension is $d\times d$ at the beginning where $d$ is the dimensionality of data, and becomes $d_{reduced}\times d$ after choosing top vectors) and then your test data $D$ ($n\times d$) is just multiplied to that matrix, you get test data in the reduced space ($D^{'}$):

$$D_{n\times d}\times E^{T} = D^{'}_{n\times d_{reduced}}$$

where the dimensionality of $E^{T}$ is $d\times d_{reduced}$ as $T$ denotes matrix transpose (you mentioned inverse which is wrong).

NOTE: Depending on how you arrange samples in your data matrix, the matrix product will be totally different. Do not get confused if you see different things in literature. The standard form of data is usually $n_{samples}\times n_{features}$ which was assumed above as well. Each row is a sample and each column is a dimension.

I hope it helped. You can comment if you had any questions.