Idempotent Matrix¶

Definition¶

An idempotent matrix, or projection matrix, is a real square matrix $\boldsymbol{P}$ satisfying $\boldsymbol{P}^2=\boldsymbol{P}$$\mathbf{P}$.

Such a matrix projects onto its range: it maps every vector into the subspace $\operatorname{range}(\boldsymbol{P})$ and leaves every vector of that subspace unchanged. A projection matrix that is also symmetric, $\boldsymbol{P}^{\top}=\boldsymbol{P}$$\mathbf{P}$, is an orthogonal projection matrix; the others are oblique projections. In statistics and econometrics "projection matrix" usually means the orthogonal case, such as the hat matrix $\boldsymbol{X}(\boldsymbol{X}^{\top}\boldsymbol{X})^{-1}\boldsymbol{X}^{\top}$ of least squares, for $\boldsymbol{X}$ of full column rank.

Interpretation¶

The square $\boldsymbol{P}^2$ is a matrix product, representing two successive applications of the same linear map. Idempotence says this composition equals one application.

Applying an idempotent matrix twice has the same effect as applying it once: projecting a vector that already lies in the subspace leaves it where it is.

In the figure, $\boldsymbol{P}^2=\boldsymbol{P}$ gives $\boldsymbol{P}(\boldsymbol{P}\boldsymbol{v})=\boldsymbol{P}\boldsymbol{v}$$\mathbf{P}\mathbf{v}$, so vectors of $\operatorname{range}\boldsymbol{P}$ stay fixed ($\mathbf{P}\mathbf{w}=$$\boldsymbol{P}\boldsymbol{w}=\boldsymbol{w}$$\mathbf{w}$) and vectors of $\ker\boldsymbol{P}$ are sent to $\boldsymbol{0}$ ($\mathbf{P}\mathbf{k}=$$\boldsymbol{P}\boldsymbol{k}=\boldsymbol{0}$$0$). Every $\boldsymbol{v}$ splits as $\boldsymbol{w}+\boldsymbol{k}$ with $\boldsymbol{w}\in\operatorname{range}\boldsymbol{P}$ and $\boldsymbol{k}\in\ker\boldsymbol{P}$ (a direct sum), and $\boldsymbol{P}\boldsymbol{v}=\boldsymbol{w}$$\mathbf{w}$. The only possible eigenvalues are $0$ and $1$; the projection need not be orthogonal (here it is oblique).

Key Results¶

• The zero square matrix and identity matrix are idempotent.
• If an idempotent matrix is invertible, then it equals the identity matrix.
• If $\boldsymbol{P}$ is idempotent, then every vector splits as $\boldsymbol{v}=\boldsymbol{P}\boldsymbol{v}+(\boldsymbol{v}-\boldsymbol{P}\boldsymbol{v})$ with $\boldsymbol{P}\boldsymbol{v}\in\operatorname{range}(\boldsymbol{P})$ and $\boldsymbol{v}-\boldsymbol{P}\boldsymbol{v}\in\ker(\boldsymbol{P})$$\ker (\mathbf{P})$, so the space is the direct sum $\operatorname{range}(\boldsymbol{P})\oplus\ker(\boldsymbol{P})$$\ker (\mathbf{P})$: $\boldsymbol{P}$ projects onto its range along its kernel.
• If $\boldsymbol{P}$ is idempotent, so is $\boldsymbol{I}-\boldsymbol{P}$$\mathbf{P}$; it projects onto $\ker(\boldsymbol{P})$ along $\operatorname{range}(\boldsymbol{P})$.
• The only possible eigenvalues of an idempotent matrix are $0$ and $1$.
• An orthogonal projection matrix is symmetric and idempotent; then $\boldsymbol{P}\boldsymbol{v}$ is the point of $\operatorname{range}(\boldsymbol{P})$ closest to $\boldsymbol{v}$, and $\boldsymbol{v}-\boldsymbol{P}\boldsymbol{v}$ is orthogonal to $\operatorname{range}(\boldsymbol{P})$.

Examples¶

• $\boldsymbol{I}_n^2=\boldsymbol{I}_n$${\mathbf{I}}_n$, so $\boldsymbol{I}_n$ is idempotent: it projects every vector onto the whole space.
• $\begin{pmatrix}1&0\\0&0\end{pmatrix}$ is idempotent and symmetric: the orthogonal projection onto the $x$-axis.
• $\begin{pmatrix}1&1\\0&0\end{pmatrix}$ is idempotent but not symmetric: an oblique projection onto the $x$-axis along the line spanned by $(1,-1)$.

Prerequisites¶

• Matrix
• Matrix Product

Related Concepts¶

• Linear Operator
• Eigenvalue and Eigenvector
• Diagonalizable Matrix

Sources¶

• Math II: Linear Algebra · Week 5

Aliases: projection matrix, projector Keywords: projection, idempotent linear operator, hat matrix, projection operator, orthogonal projection, orthogonal projector, oblique projection Symbols: element of

Last updated: 27 October 2026 11:52