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 of that vector.

Definition

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

$C_{n}=I_{n}-{\tfrac {1}{n}}J_{n}$

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

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

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

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

$C_{n}\,$ is an orthogonal 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 $J_{n,1}$. (This is the subspace of all n-vectors whose components sum to zero.)

The trace of $C_{n}$ is $n(n-1)/n=n-1$.

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 $X$.

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

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

$S=X\,C_{n}(X\,C_{n})^{\mathrm {T} }=X\,C_{n}\,C_{n}\,X\,^{\mathrm {T} }=X\,C_{n}\,X\,^{\mathrm {T} }.$

$C_{n}$ is the covariance matrix of the multinomial distribution, in the special case where the parameters of that distribution are $k=n$, and $p_{1}=p_{2}=\cdots =p_{n}={\frac {1}{n}}$.