Centering matrix

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}X J_{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}. == References ==