Centering matrix

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

$C_n = I_n - \tfrac{1}{n}\mathbb{O}$

where $I_n\,$ is the identity matrix of size n and $\mathbb{O}$ is an n-by-n matrix of all 1's. This can also be written as:

$C_n = I_n - \tfrac{1}{n}\mathbf{1}\mathbf{1}^\top$

where $\mathbf{1}$ is the column-vector of n ones and where $\top$ denotes matrix transpose.

For example

$C_1 = \begin{bmatrix} 0 \end{bmatrix}$,

$C_2= \left[ \begin{array}{rrr} 1 & 0 \\ \\ 0 & 1 \end{array} \right] - \frac{1}{2}\left[ \begin{array}{rrr} 1 & 1 \\ \\ 1 & 1 \end{array} \right] = \left[ \begin{array}{rrr} \frac{1}{2} & -\frac{1}{2} \\ \\ -\frac{1}{2} & \frac{1}{2} \end{array} \right]$ ,

$C_3 = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ \\ 0 & 1 & 0 \\ \\ 0 & 0 & 1 \end{array} \right] - \frac{1}{3}\left[ \begin{array}{rrr} 1 & 1 & 1 \\ \\ 1 & 1 & 1 \\ \\ 1 & 1 & 1 \end{array} \right] = \left[ \begin{array}{rrr} \frac{2}{3} & -\frac{1}{3} & -\frac{1}{3} \\ \\ -\frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \\ \\ -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \end{array} \right]$

Properties

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

$C_n\,\mathbf{v} = \mathbf{v}-(\tfrac{1}{n}\mathbf{1}'\mathbf{v})\mathbf{1}$

where $\tfrac{1}{n}\mathbf{1}'\mathbf{v}$ 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 the mean has been removed, 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 eigenvalue 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, $C_n\mathbf{v}$ 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, $S=(X-\mu\mathbf{1}')(X-\mu\mathbf{1}')'$ of a data sample $X\,$, where $\mu=\tfrac{1}{n}X\mathbf{1}$ is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as

$S=X\,C_n(X\,C_n)'=X\,C_n\,C_n\,X\,'=X\,C_n\,X\,'.$

$C_n$ is the covariance matrix of the multinomial distribution, in the special case where the parameters of that distribution are $k=n$, and $p_1=p_2=\cdots=p_n=\frac{1}{n}$.