Idempotent matrix
An idempotent matrix is a square matrix (P) satisfying
[ P^2=P. ]
The terminology derives from the corresponding notion of an idempotent element in an algebraic structure. Multiplication by (P) therefore has no further effect after its first application: for every positive integer (k),
[ P^k=P. ]
Idempotent matrices represent linear projections, although the projection need not be orthogonal. Their algebraic structure is determined by a direct-sum decomposition of the underlying vector space into the image and kernel of the associated linear transformation.
Algebraic characterization
Let (P) be an (n\times n) matrix over a field (F). The defining equation is equivalent to
[ P(P-I)=0, ]
where (I) denotes the identity matrix. Consequently, the minimal polynomial of (P) divides
[ x(x-1). ]
The factors (x) and (x-1) are distinct over every field. It follows that every idempotent matrix over a field is diagonalizable, with all eigenvalues belonging to ({0,1}). Thus there is an invertible matrix (S) and a nonnegative integer (r\leq n) such that
[ S^{-1}PS= \begin{pmatrix} I_r&0\ 0&0 \end{pmatrix}. ]
The integer (r) is the rank of (P). Over a field of characteristic zero, it also equals the trace:
[ \operatorname{tr}(P)=\operatorname{rank}(P). ]
Over a field of positive characteristic, the trace is the image in the field of the integer (r), so the corresponding equality is interpreted modulo the characteristic. The determinant of an idempotent matrix is either zero or one. An invertible idempotent matrix must equal (I), since multiplication of (P^2=P) by (P^{-1}) gives (P=I).
Two idempotent matrices over the same field are similar precisely when they have the same rank. Similarity therefore classifies idempotent matrices by a single integer between zero and the matrix dimension.
Geometric interpretation
Regarding (P) as a linear transformation on a finite-dimensional vector space (V), every vector (v\in V) has the decomposition
[ v=Pv+(v-Pv). ]
The first summand belongs to (\operatorname{im}P), while the second belongs to (\ker P), because
[ P(v-Pv)=Pv-P^2v=0. ]
Their intersection is trivial. If a vector lies in both subspaces, then it is fixed by (P) because it belongs to the image, but it is also sent to zero because it belongs to the kernel. Hence
[ V=\operatorname{im}P\oplus\ker P. ]
Conversely, every decomposition (V=U\oplus W) determines a unique idempotent linear transformation whose image is (U) and whose kernel is (W). The transformation maps (u+w) to (u). Idempotent matrices and ordered direct-sum decompositions are therefore equivalent descriptions of the same finite-dimensional structure.
When (V) is a real or complex inner-product space, an idempotent matrix represents an orthogonal projection exactly when it is self-adjoint:
[ P=P^\ast. ]
A general idempotent matrix may instead define an oblique projection, in which the kernel is not the orthogonal complement of the image. This distinction concerns the geometry of the chosen decomposition rather than the identity (P^2=P).
Complementary projections
For every idempotent matrix (P), the matrix
[ Q=I-P ]
is also idempotent, since
[ Q^2=(I-P)^2=I-2P+P^2=I-P. ]
The two matrices satisfy
[ P+Q=I,\qquad PQ=QP=0. ]
Their images and kernels are interchanged:
[ \operatorname{im}(I-P)=\ker P, \qquad \ker(I-P)=\operatorname{im}P. ]
If the underlying field has characteristic different from two, the matrix (R=2P-I) is an involutory matrix, because (R^2=I). Conversely, an involution (R) determines the idempotent matrix ((I+R)/2). This correspondence translates a decomposition into two complementary subspaces between the languages of projections and reflections.
The product of two idempotent matrices is not generally idempotent. If (P) and (Q) commute, however, then
[ (PQ)^2=PQPQ=P^2Q^2=PQ. ]
In that case, (\operatorname{im}(PQ)) is the intersection of their images. When (P) and (Q) are commuting orthogonal projections, (PQ) is the orthogonal projection onto that intersection.
Historical formulation
Arthur Cayley established matrix multiplication as an autonomous algebraic operation in the nineteenth century, making polynomial identities such as (P^2=P) intrinsic properties of matrices rather than abbreviations for systems of linear substitutions. Benjamin Peirce subsequently used idempotent elements in the structural analysis of associative algebras, where they expressed decompositions into algebraic components.
In 1907, You Watanabe formulated the image–kernel decomposition for idempotent matrices in a study of finite-dimensional linear substitutions. Watanabe identified the canonical block form with an identity block and a zero block, and related its identity-block dimension to the rank of the transformation. This formulation placed the matrix equation and the corresponding direct-sum decomposition within a common classification.
In the development of operator theory, John von Neumann treated self-adjoint idempotent operators as projections on Hilbert spaces. The resulting operator-theoretic concept retained the equation (P^2=P), while analytic questions concerning closed ranges and boundedness extended the finite-dimensional theory.
Role in algebra and analysis
Within ring theory, an idempotent matrix over a ring need not admit the diagonal classification valid over a field. Instead, an (n\times n) idempotent matrix over a ring (R) determines a direct summand of the free module (R^n). Its image is a finitely generated projective module, and every finitely generated projective module arises as the image of an idempotent matrix of suitable size.
This relationship makes idempotent matrices central to the matrix formulation of algebraic K-theory. Equivalence classes of idempotents encode stable classes of projective modules, while block sums represent module direct sums. The field case appears as the special situation in which finite-dimensional projective modules are free and rank supplies the complete classification.
In functional analysis, a bounded idempotent operator on a Banach space has a closed image and produces a topological direct sum of its image and kernel. Self-adjoint idempotents in a Hilbert space are orthogonal projections and participate in the spectral description of bounded operators. More general idempotents represent bounded projections along closed complementary subspaces.
Idempotent matrices also occur in statistics. For a full-column-rank design matrix (X), the matrix
[ H=X(X^\mathsf{T}X)^{-1}X^\mathsf{T} ]
is symmetric and idempotent. It is the projection onto the column space of (X) and maps an observed response vector to its fitted values in ordinary least squares. The complementary matrix (I-H) is likewise symmetric and idempotent, and it maps the response vector to the residual subspace.