Matrix that, squared, equals itself
In
linear algebra, an idempotent matrix is a
matrix which, when multiplied by itself, yields itself.
[1]
[2] 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.
Example
Examples of idempotent matrices are:
Examples of idempotent matrices are:
Real 2 × 2 case
If a matrix is idempotent, then
- implying so or
- implying so or
Thus, a necessary condition for a matrix to be idempotent is that either it is
diagonal or its
trace equals 1.
For idempotent diagonal matrices, and must be either 1 or 0.
If , the matrix will be idempotent provided so a satisfies the
quadratic equation
- or
which is a
circle with center (1/2, 0) and radius 1/2. In terms of an angle θ,
- is idempotent.
However, is not a necessary condition: any matrix
- with 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).
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.
Eigenvalues
An idempotent matrix is always
diagonalizable.
[3] 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.
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
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: .
Applications
Idempotent matrices arise frequently in
regression analysis and
econometrics. For example, in
ordinary least squares, the regression problem is to choose a vector β of coefficient estimates so as to minimize the sum of squared residuals (mispredictions) ei: in matrix form,
- Minimize
where is a vector of
dependent variable observations, and 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
[2]
Here both and (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:
The idempotency of plays a role in other calculations as well, such as in determining the variance of the estimator .
An idempotent linear operator 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
References