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. The centering matrix of size n is defined as the n-by-n matrix where is the identity matrix of size n and is an n-by-n matrix of all 1's. For example Given a column-vector, of size n, the centering property of can be expressed as where is a column vector of ones and is the mean of the components of . is symmetric positive semi-definite. is idempotent, so that , for . Once the mean has been removed, it is zero and removing it again has no effect. is singular. The effects of applying the transformation cannot be reversed. has the eigenvalue 1 of multiplicity n − 1 and eigenvalue 0 of multiplicity 1. has a nullspace of dimension 1, along the vector . is an orthogonal projection matrix. That is, is a projection of onto the (n − 1)-dimensional subspace that is orthogonal to the nullspace . (This is the subspace of all n-vectors whose components sum to zero.) The trace of is . 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 . The left multiplication by subtracts a corresponding mean value from each of the n columns, so that each column of the product has a zero mean. Similarly, the multiplication by on the right subtracts a corresponding mean value from each of the m rows, and each row of the product has a zero mean. The multiplication on both sides creates a doubly centred matrix , whose row and column means are equal to zero. The centering matrix provides in particular a succinct way to express the scatter matrix, of a data sample , where is the sample mean.