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# 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
$C_n = I_n - frac\left\{1\right\}\left\{n\right\}mathbf\left\{1\right\}mathbf\left\{1\right\}\text{'}$
where $I_n,$ is the identity matrix of size n, $mathbf\left\{1\right\}$ is the column-vector of n ones and where $\left\{,\right\}\text{'}$ denotes matrix transpose. For example

$C_1 = begin\left\{bmatrix\right\}$
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, $mathbf\left\{v\right\},$ of size n, the centering property of $C_n,$ can be expressed as
$C_n,mathbf\left\{v\right\} = mathbf\left\{v\right\}-\left(frac\left\{1\right\}\left\{n\right\}mathbf\left\{1\right\}\text{'}mathbf\left\{v\right\}\right)mathbf\left\{1\right\}$
where $frac\left\{1\right\}\left\{n\right\}mathbf\left\{1\right\}\text{'}mathbf\left\{v\right\}$ is the mean of the components of $mathbf\left\{v\right\},$.

$C_n,$ 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 $C_n,mathbf\left\{v\right\}$ 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 $mathbf\left\{1\right\}$.

$C_n,$ is a projection matrix. That is, $C_nmathbf\left\{v\right\}$ is a projection of $mathbf\left\{v\right\},$ onto the (n − 1)-dimensional subspace that is orthogonal to the nullspace $mathbf\left\{1\right\}$. (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, $S=\left(X-mumathbf\left\{1\right\}\text{'}\right)\left(X-mumathbf\left\{1\right\}\text{'}\right)\text{'}$ of a data sample $X,$, where $mu=tfrac\left\{1\right\}\left\{n\right\}Xmathbf\left\{1\right\}$ is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as

$S=X,C_n\left(X,C_n\right)\text{'}=X,C_n,C_n,X,\text{'}=X,C_n,X,\text{'}.$

## References

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