# Rank of centered data

Consider a ${n\times p}$ matrix ${X}$. In general, this matrix has rank

$\displaystyle \mathrm{rank}(X)\leq \min\{n,p\}.$

Now, suppose we wish to column-center the data. We can do this algebraically by using what is known as the centering matrix,

$\displaystyle H_n=I_n-\frac{1}{n}\mathbf{O},$

where ${\mathbf{O}}$ is the matrix where ${\mathbf{O}_{ij}=1}$ for all ${i,j}$. Multiplying ${H_n}$ with ${X}$ results to the centering of all the columns.

The vector of ones is the only independent element in its nullspace and so ${\mathrm{rank}(H_n)=n-1}$. Therefore,

$\displaystyle \mathrm{rank}(H_nX)\leq \min\{n-1,n,p\}=\min\{n-1,p\}.$

Similarly, for the row-centered matrix ${XH_p}$,

$\displaystyle \mathrm{rank}(XH_p)\leq \min\{p-1,n\}.$