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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.

Definition

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

$$C_1 = \begin{bmatrix} 0 \end{bmatrix} ,$$

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

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

Properties

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

$$C_n\,\mathbf{v} = \mathbf{v}-(\tfrac{1}{n}\mathbf{1}'\mathbf{v})\mathbf{1}$$

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.)

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, $$S=(X-\mu\mathbf{1}')(X-\mu\mathbf{1}')'$$ 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 \( S=X\,C_n(X\,C_n)'=X\,C_n\,C_n\,X\,'=X\,C_n\,X\,'.$$

$$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}.$$

References

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

Mathematics Encyclopedia