In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and multivariate statistics

Multivariate statistics

Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...

, the centering matrix is a symmetric and idempotent matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, which when multiplied with a vector has the same effect as subtracting the mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 of the components of the vector from every component.

Definition

The centering matrix of size n is defined as the n-by-n matrix
where is the identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

 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 (n − 1)-dimensional subspace

Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

 that is orthogonal to the nullspace . (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 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.

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

Scatter matrix

In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix of the multivariate normal distribution.-Definition:...

, of a data sample , where is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as