Centering matrix

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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.


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]


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.)


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


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}.


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