# Centering matrix

Short description: Kind of matrix

In mathematics and multivariate statistics, the centering matrix[1] 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 of that vector.

## Definition

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

$\displaystyle{ C_n = I_n - \tfrac{1}{n}J_n }$

where $\displaystyle{ I_n\, }$ is the identity matrix of size n and $\displaystyle{ J_n }$ is an n-by-n matrix of all 1's.

For example

$\displaystyle{ C_1 = \begin{bmatrix} 0 \end{bmatrix} }$,
$\displaystyle{ C_2= \left[ \begin{array}{rrr} 1 & 0 \\ 0 & 1 \end{array} \right] - \frac{1}{2}\left[ \begin{array}{rrr} 1 & 1 \\ 1 & 1 \end{array} \right] = \left[ \begin{array}{rrr} \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} \end{array} \right] }$ ,
$\displaystyle{ C_3 = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] - \frac{1}{3}\left[ \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right] = \left[ \begin{array}{rrr} \frac{2}{3} & -\frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \end{array} \right] }$

## Properties

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

$\displaystyle{ C_n\,\mathbf{v} = \mathbf{v} - (\tfrac{1}{n}J_{n,1}^\textrm{T}\mathbf{v})J_{n,1} }$

where $\displaystyle{ J_{n,1} }$ is a column vector of ones and $\displaystyle{ \tfrac{1}{n}J_{n,1}^\textrm{T}\mathbf{v} }$ is the mean of the components of $\displaystyle{ \mathbf{v}\, }$.

$\displaystyle{ C_n\, }$ is symmetric positive semi-definite.

$\displaystyle{ C_n\, }$ is idempotent, so that $\displaystyle{ C_n^k=C_n }$, for $\displaystyle{ k=1,2,\ldots }$. Once the mean has been removed, it is zero and removing it again has no effect.

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

$\displaystyle{ C_n\, }$ has the eigenvalue 1 of multiplicity n − 1 and eigenvalue 0 of multiplicity 1.

$\displaystyle{ C_n\, }$ has a nullspace of dimension 1, along the vector $\displaystyle{ J_{n,1} }$.

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

The trace of $\displaystyle{ C_n }$ is $\displaystyle{ n(n-1)/n = n-1 }$.

## Application

Although multiplication by the centering matrix is not a computationally efficient way of removing the mean from a vector, it is a convenient analytical tool. 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 an m-by-n matrix $\displaystyle{ X }$.

The left multiplication by $\displaystyle{ C_m }$ subtracts a corresponding mean value from each of the n columns, so that each column of the product $\displaystyle{ C_m\,X }$ has a zero mean. Similarly, the multiplication by $\displaystyle{ C_n }$ on the right subtracts a corresponding mean value from each of the m rows, and each row of the product $\displaystyle{ X\,C_n }$ has a zero mean. The multiplication on both sides creates a doubly centred matrix $\displaystyle{ C_m\,X\,C_n }$, whose row and column means are equal to zero.

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

$\displaystyle{ S=X\,C_n(X\,C_n)^{\mathrm{T}}=X\,C_n\,C_n\,X\,^{\mathrm{T}}=X\,C_n\,X\,^{\mathrm{T}}. }$

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

## References

1. John I. Marden, Analyzing and Modeling Rank Data, Chapman & Hall, 1995, ISBN:0-412-99521-2, page 59.