Idempotent matrix

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. Examples of idempotent matrices are: Examples of idempotent matrices are: 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. 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.