2-EXPTIME

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

$C_n=I_n-\frac{1}{n}\mathbb{𝕆}$

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:

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

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

For example

$C_1=\left[\begin{array}{c}0\end{array}\right]$,

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

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

Properties

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

$C_n\mathbf{v}=\mathbf{v}-(\frac{1}{n}{\mathbf{1}}^{\prime }\mathbf{v})\mathbf{1}$

where $\frac{1}{n}{\mathbf{1}}^{\prime }\mathbf{v}$ 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 $k=1,2,\dots$. 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, $S=(X-\mu {\mathbf{1}}^{\prime })(X-\mu {\mathbf{1}}^{\prime }{)}^{\prime }$ of a data sample $X$, where $\mu =\frac{1}{n}X\mathbf{1}$ is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as

$S=X{C}_n(X{C}_n{)}^{\prime }=X{C}_n{C}_{n}X{}^{\prime }=X{C}_{n}X{}^{\prime }.$

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