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
where is the identity matrix of size n and is an n-by-n matrix of all 1's. This can also be written as:
where is the column-vector of n ones and where denotes matrix transpose.
For example

Properties

Given a column-vector, of size n, the centering property of can be expressed as
where 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 a projection matrix. That is, is a projection of onto the -dimensional subspace that is orthogonal to the nullspace.

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, the multiplication removes the means from each of the n columns, while removes the means from each of the m rows.
For an n-by-n matrix, the multiplication creates a doubly centred matrix where both row and column means are equal to zero. Hence:.
The centering matrix provides in particular a succinct way to express the scatter matrix, of a data sample, where is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as
is the covariance matrix of the multinomial distribution, in the special case where the parameters of that distribution are, and.