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Idempotent Matrix

A square matrix that, multiplied by itself, returns itself unchanged -- the algebraic signature of a projection, and the property (alongside symmetry) that identifies the regression hat matrix as an orthogonal projection onto the column space of the predictors.

Linear Algebra for StatisticsAdvancedFree

In Plain English

An idempotent matrix is one where applying it twice does nothing more than applying it once, the matrix version of the fact that projecting a point onto a wall, then projecting that same result onto the wall again, lands you in exactly the same spot both times. This 'do it again, nothing changes' property is exactly what a projection is, geometrically, which is why the regression hat matrix, an idempotent, symmetric matrix, is precisely the projection that produces fitted values from observed data.

Definition

A square matrix $A$ is idempotent if $A^{2} = AA = A$ , applying it a second time has no further effect beyond applying it once. A symmetric idempotent matrix is, geometrically, exactly an orthogonal projection matrix onto some subspace, and the regression hat matrix $H = X (X^{⊤} X)^{- 1} X^{⊤}$ is the canonical statistical example, both symmetric and idempotent, projecting the response vector $y$ orthogonally onto the column space spanned by the predictors to produce the fitted values $\hat{y}$ . A key consequence used throughout regression theory is that an idempotent matrix's eigenvalues are all either 0 or 1, and its trace (sum of eigenvalues) equals its rank, which is why $trace (H)$ exactly equals the number of estimated regression parameters.

Formula

A^{2} = A

Notation

A^{2}

—A multiplied by itself, A times A

Properties

An idempotent matrix's eigenvalues can only be 0 or 1 -- this follows directly from $A^{2} = A$ applied to an eigenvector, $λ^{2} = λ$ has only the solutions $λ = 0$ or $λ = 1$ , and is the reason an idempotent matrix's trace (the sum of its eigenvalues) always equals an integer, specifically its rank.
Symmetry combined with idempotency exactly characterizes an orthogonal projection matrix -- the regression hat matrix satisfies both properties, which is the formal algebraic reason it can be interpreted geometrically as projecting the observed response vector onto the subspace spanned by the model's predictors.
If $P$ is idempotent, then $I - P$ is also idempotent, and projects onto the orthogonal complementary subspace -- in regression, $I - H$ is the idempotent matrix that produces the residual vector, $(I - H) y = y - \hat{y}$ , the projection onto the space orthogonal to the fitted values.

At a Glance

Defining propertyA^2 = A -- applying it twice equals applying it once

Eigenvalues are always0 or 1, nothing else

Trace equalsrank of the matrix

Symmetric + idempotentexactly characterizes an orthogonal projection matrix

Last updated September 8, 2026← Back to the Dictionary