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 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.
of size n, the centering property of
is a column vector of ones and
is the mean of the components of
is symmetric positive semi-definite.
Once the mean has been removed, it is zero and removing it again has no effect.
The effects of applying the transformation
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.
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.)
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
Similarly, the multiplication by
on the right subtracts a corresponding mean value from each of the m rows, and each row of the product
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
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