Projection (linear algebra)

Lore & Background

A projection on a vector space V is a linear operator P: V → V such that P² = P. When V has an inner product and is complete (a Hilbert space), a projection is called orthogonal if it satisfies ⟨Px, y⟩ = ⟨x, Py⟩ for all x, y ∈ V; otherwise it is called oblique. A square matrix P is a projection matrix if P² = P. An orthogonal projection matrix satisfies P² = P = Pᵀ for real matrices or P² = P = P* for complex matrices, where Pᵀ is the transpose and P* is the adjoint or Hermitian transpose. A projection matrix that is not orthogonal is called oblique. The eigenvalues of a projection matrix must be 0 or 1.

Reader's Guide

The concept of projection is fundamental in linear algebra and functional analysis, providing a rigorous mathematical framework for the intuitive idea of projecting points onto a subspace. The definition via idempotence (P² = P) is simple yet powerful, leading to a rich theory. In Hilbert spaces, the distinction between orthogonal and oblique projections is crucial: orthogonal projections preserve inner products and correspond to the closest-point mapping, while oblique projections do not. Projection matrices, which satisfy P² = P, are widely used in applications such as computer graphics, statistics (e.g., regression), and signal processing. The fact that eigenvalues of a projection matrix are only 0 or 1 reflects its nature as a 'selector' of a subspace. The source article provides concrete examples, such as the orthogonal projection onto the xy-plane in ℝ³, represented by a diagonal matrix with ones for the first two coordinates and zero for the third, and an oblique projection example with a non-symmetric matrix. These examples illustrate how projections can be represented and computed in practice.

Did You Know?

• A projection is a linear transformation P such that P∘P = P, meaning it is idempotent.
• An orthogonal projection on a Hilbert space satisfies ⟨Px, y⟩ = ⟨x, Py⟩ for all vectors x, y.
• The eigenvalues of a projection matrix must be either 0 or 1.
• A projection matrix that is not orthogonal is called an oblique projection matrix.

The Idempotent Heart of Projection

A projection, in the language of linear algebra and functional analysis, is fundamentally a linear map that sends a vector space back onto itself while satisfying a single, elegant constraint: applying the map twice yields exactly the same outcome as applying it once. This property, known as idempotency, is expressed as P∘P = P, and it is the defining characteristic that separates a projection from an arbitrary endomorphism. Because the second application changes nothing, the transformation effectively settles every vector into a subspace and then leaves that subspace untouched thereafter. In other words, the image of the projection is a fixed set of points; any vector already lying in that image is mapped to itself. This algebraic structure is not an abstract curiosity—it is a rigorous generalization of the everyday geometric idea of casting a shadow or flattening a three-dimensional figure onto a plane. When one wishes to understand how a projection deforms an entire geometrical object, the standard approach is to track what happens to each individual point within that object, since the linear nature of the map guarantees that the collective behavior is fully determined by the pointwise action.

Orthogonal and Oblique: Two Kinds of Projection

When the underlying vector space carries an inner product and is complete—making it a Hilbert space—the notion of perpendicularity enters the picture and splits projections into two distinct families. A projection P on such a space is classified as orthogonal if it commutes with the inner product in a very specific way: for every pair of vectors x and y, the inner product of Px with y equals the inner product of x with Py. This symmetry condition encodes the geometric intuition that the drop from a point to its projected image is perpendicular to the target subspace. Any projection that fails to satisfy this symmetry is termed an oblique projection, meaning the mapping tilts vectors at an angle rather than sending them straight onto the subspace. The distinction is not merely cosmetic; it governs whether the projection preserves angles and lengths in the projected direction, and it determines which algebraic identities the operator obeys. In practical terms, orthogonal projections are the ones most often encountered in approximation theory and signal processing, while oblique projections arise when one deliberately projects along a non-perpendicular direction to isolate a particular component of a vector.

The Matrix View and the Binary Eigenvalue Rule

When a projection is represented as a square matrix acting on a finite-dimensional space, the idempotency condition becomes the compact equation P² = P. A matrix satisfying this identity is called a projection matrix. If, in addition, the matrix equals its own transpose (in the real case) or its own adjoint, also known as the Hermitian transpose (in the complex case), then it is an orthogonal projection matrix, written as P² = P = P^T or P² = P = P*. A projection matrix that does not meet this symmetry requirement is labeled an oblique projection matrix. One of the most striking consequences of idempotency is a severe restriction on the spectrum: every eigenvalue of a projection matrix must be either 0 or 1. This binary constraint follows directly from the equation P² = P, because if Pv = λv for some eigenvector v, then substituting into the idempotency relation forces λ² = λ, which admits only the two solutions. The eigenspace associated with the eigenvalue 1 spans the image of the projection, while the eigenspace for 0 spans its kernel, giving a clean decomposition of the entire space into the part that is preserved and the part that is annihilated.

Flattening Space: The xy-Plane Example

A particularly transparent illustration of an orthogonal projection lives in three-dimensional Euclidean space. Consider the map that takes any point (x, y, z) and sends it to (x, y, 0), effectively erasing the z-coordinate and dropping the point straight down onto the xy-plane. This operation is represented by the 3×3 matrix with ones on the first two diagonal entries and zeros everywhere else: the first row is [1, 0, 0], the second is [0, 1, 0], and the third is [0, 0, 0]. Multiplying this matrix by the column vector [x, y, z] produces exactly [x, y, 0], confirming that the z-component is discarded while the x and y components pass through unchanged. To verify that this matrix truly qualifies as a projection, one simply multiplies it by itself and observes that the result is identical to the original matrix, satisfying P² = P. Moreover, because the matrix is symmetric (equal to its own transpose), it meets the additional criterion for an orthogonal projection matrix. The geometric picture is immediate: every point in R³ is projected perpendicularly onto the horizontal plane, and any point already sitting in that plane remains exactly where it was.