In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. to be defined must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elementss of matrix rings.

Example

Real 2 × 2 case

is idempotent, then

matrix to be idempotent is that either it is diagonal or its trace must be either 1 or 0.

so a satisfies the quadratic equation

which is a circle with center (1/2, 0) and radius 1/2. In terms of an angle ,

1. is idempotent.

is not a necessary condition: any matrix

1. is idempotent.

Properties

Singularity and regularity

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).

, assuming that .

When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds since

If a matrix , since is idempotent. Hence by the principle of induction, the result follows.

Eigenvalues

An idempotent matrix is always diagonalizable. Its eigenvalues 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.

Trace

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).

Relationships between idempotent matrices

. 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.

Applications

Idempotent matrices arise frequently in regression analysis and econometrics. For example, in ordinary least squares, the regression problem is to choose a vector of coefficient estimates so as to minimize the sum of squared residuals (mispredictions) ei: in matrix form,

is a vector of dependent variable 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

(the latter being known as the hat matrix) are idempotent and symmetric matrices, a fact which allows simplification when the sum of squared residuals is computed:

.

is a projection operator on the range space along its null space is an orthogonal projection operator if and only if it is idempotent and symmetric.

See also

• Idempotence

• Nilpotent

• Projection (linear algebra)

• Hat matrix

References