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

is the identity matrix of size n is an n-by-n matrix of all 1's.

For example

1. ,

   ,

Properties

of size n, the centering property can be expressed as

is a column vector of ones.

is symmetric positive semi-definite.

is idempotent. Once the mean has been removed, it is zero and removing it again has no effect.

is singular cannot be reversed.

has the eigenvalue 1 of multiplicity n1 and eigenvalue 0 of multiplicity 1.

has a nullspace.

is an orthogonal projection matrix onto the (n1)-dimensional subspace. (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 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.

subtracts a corresponding mean value from each of the n on the right subtracts a corresponding mean value from each of the m, whose row and column means are equal to zero.

The centering matrix provides in particular a succinct way to express the scatter matrix 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.

References