In mathematics and multivariate statistics, the centering matrix is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component.

Definition

The centering matrix of size n is defined as the n-by-n matrix

$C\_n\; =\; I\_n\; -\; frac\{1\}\{n\}mathbf\{1\}mathbf\{1\}\text{'}$

where $I\_n,$ is the identity matrix of size n, $mathbf\{1\}$ is the column-vector of n ones and where $\{,\}\text{'}$ denotes matrix transpose. For example

$C\_1\; =\; begin\{bmatrix\}$

0 end{bmatrix} , C_2 = left[begin{array}{rrr} frac{1}{2} & -frac{1}{2} -frac{1}{2} & frac{1}{2} end{array} right] , C_3 = left[begin{array}{rrr} frac{2}{3} & -frac{1}{3} & -frac{1}{3} -frac{1}{3} & frac{2}{3} & -frac{1}{3} -frac{1}{3} & -frac{1}{3} & frac{2}{3} end{array} right]

Properties

Given a column-vector, $mathbf\{v\},$ of size n, the centering property of $C\_n,$ can be expressed as

$C\_n,mathbf\{v\}\; =\; mathbf\{v\}-(frac\{1\}\{n\}mathbf\{1\}\text{'}mathbf\{v\})mathbf\{1\}$

where $frac\{1\}\{n\}mathbf\{1\}\text{'}mathbf\{v\}$ is the mean of the components of $mathbf\{v\},$.

$C\_n,$ is symmetric positive semi-definite.

$C\_n,$ is idempotent, so that $C\_n^k=C\_n$, for $k=1,2,ldots$. Once you have removed the mean, it is zero and removing it again has no effect.

$C\_n,$ is singular. The effects of applying the transformation $C\_n,mathbf\{v\}$ cannot be reversed.

$C\_n,$ has the eigenvalue 1 of multiplicity n − 1 and 0 of multiplicity 1.

$C\_n,$ has a nullspace of dimension 1, along the vector $mathbf\{1\}$.

$C\_n,$ is a projection matrix. That is, $C\_nmathbf\{v\}$ is a projection of $mathbf\{v\},$ onto the (n − 1)-dimensional subspace that is orthogonal to the nullspace $mathbf\{1\}$. (This is the subspace of all n-vectors whose components sum to zero.)

Application

Although multiplication by the centering matrix is not a computationally efficient way of removing the mean from a vector, it forms an analytical tool that conveniently and succinctly expresses mean removal. It can be used not only to remove the mean of a single vector, but also of multiple vectors stored in the rows or columns of a matrix. For an m-by-n matrix $X,$, the multiplication $C\_m,X$ removes the means from each of the n columns, while $X,C\_n$ removes the means from each of the m rows.

The centering matrix provides in particular a succinct way to express the scatter matrix, $S=(X-mumathbf\{1\}\text{'})(X-mumathbf\{1\}\text{'})\text{'}$ of a data sample $X,$, where $mu=tfrac\{1\}\{n\}Xmathbf\{1\}$ is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as

$S=X,C\_n(X,C\_n)\text{'}=X,C\_n,C\_n,X,\text{'}=X,C\_n,X,\text{'}.$

References