Idempotent matrix

Matrix that, squared, equals itself

In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if For this product to be defined, must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.

The only non singular idempotent matrix is the identity matrix; that is, if a non identity matrix is idempotent, its number of independent rows (and columns) is less than its number of rows (and columns). This can be seen from writing , assuming that A has full rank (is non singular), and pre multiplying by to obtain
When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds since

If a matrix A is idempotent then for all positive integers n, This can be shown using proof by induction. Clearly we have the result for , as Suppose that Then, , since A is idempotent. Hence by the principle of induction, the result follows.

An idempotent matrix is always diagonalizable. Its eigenvalues are either 0 or 1: if is a non zero eigenvector of some idempotent matrix and its associated eigenvalue, then which implies This further implies that the determinant of an idempotent matrix is always 0 or 1. As stated above, if the determinant is equal to one, the matrix is invertible and is therefore the identity matrix.

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 statistics, for example, in establishing the degree of bias in using a sample variance as an estimate of a population variance).

In regression analysis, the matrix is known to produce the residuals from the regression of the vector of dependent variables on the matrix of covariates (See the section on Applications.) Now, let be a matrix formed from a subset of the columns of , and let It is easy to show that both and are idempotent, but a somewhat surprising fact is that This is because , or in other words, the residuals from the regression of the columns of on are 0 since can be perfectly interpolated as it is a subset of (by direct substitution it is also straightforward to show that ). This leads to two other important results: one is that is symmetric and idempotent, and the other is that , i. e., is orthogonal to These results play a key role, for example, in the derivation of the F test. Any similar matrices of an idempotent matrix are also idempotent. Idempotency is conserved under a change of basis. This can be shown through multiplication of the transformed matrix with being idempotent: .