In algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix M is idempotent if and only if MM = M. For this product MM to be defined, M must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.
Contents
Example
Examples of a
Real 2 × 2 case
If a matrix
Thus a necessary condition for a 2 × 2 matrix to be idempotent is that either it is diagonal or its trace equals 1. Notice that, for idempotent diagonal matrices,
If b = c, the matrix
which is a circle with center (1/2, 0) and radius 1/2. In terms of an angle θ,
However, b = c is not a necessary condition: any matrix
Properties
With the exception of the identity matrix, an idempotent matrix is singular; that is, its number of independent rows (and columns) is less than its number of rows (and columns). This can be seen from writing
When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds since [I − M][I − M] = I − M − M + M2 = I − M − M + M = I − M.
A matrix
An idempotent matrix is always diagonalizable and its eigenvalues are either 0 or 1. The trace of an idempotent matrix — the sum of the elements on its main diagonal — equals the rank of the matrix and thus is always an integer. This provides an easy way of computing the rank, or alternatively an easy way of determining the trace of a matrix whose elements are not specifically known (which is helpful in econometrics, for example, in establishing the degree of bias in using a sample variance as an estimate of a population variance).
Applications
Idempotent matrices arise frequently in regression analysis and econometrics. For example, in ordinary least squares, the regression problem is to choose a vector
where y is a vector of dependent variable observations, and X is a matrix each of whose columns is a column of observations on one of the independent variables. The resulting estimator is
where superscript T indicates a transpose, and the vector of residuals is
Here both M and
The idempotency of M plays a role in other calculations as well, such as in determining the variance of the estimator
An idempotent linear operator P is a projection operator on the range space R(P) along its null space N(P). P is an orthogonal projection operator if and only if it is idempotent and symmetric.