Idempotent matrix

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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.

Definition[edit]

A square matrix is called idempotent if, multiplied by itself, yields itself, i.e.

Example[edit]

Examples of idempotent matrices are:

Examples of idempotent matrices are:

Real 2 × 2 case[edit]

If a matrix is idempotent, then

  • implying so or
  • implying so or

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, 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[edit]

Singularity and regularity[edit]

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

.

A matrix A is idempotent if and only if for all positive integers n, . The 'if' direction trivially follows by taking . The 'only if' part can be shown using proof by induction. Clearly we have the result for , as . Suppose that . Then, , as required. Hence by the principle of induction, the result follows.

Eigenvalues[edit]

An idempotent matrix is always diagonalizable and its eigenvalues are either 0 or 1.[3]

Trace[edit]

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).

Applications[edit]

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[edit]

References[edit]

  1. ^ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). New York: McGraw–Hill. p. 80. ISBN 0070108137.
  2. ^ a b Greene, William H. (2003). Econometric Analysis (5th ed.). Upper Saddle River, NJ: Prentice–Hall. pp. 808–809. ISBN 0130661899.
  3. ^ Horn, Roger A.; Johnson, Charles R. (1990). Matrix analysis. Cambridge University Press. p. p. 148. ISBN 0521386322.