Projection (linear algebra)

In linear algebra, a projection is a linear transformation P such that P² = P, i.e., an idempotent transformation. A matrix is a projection if the transformation it represents is a projection. An m × m matrix projection maps an m-dimensional vector space onto a k-dimensional subspace (k ≤ m). A special class of projections is the class of orthogonal projections, which are self-adjoint projections.

One such common projection is the projection of one vector in Rⁿ onto another. For example, we can project the vector (1/2, 1/2)^T onto the vector (0, 1)^T, to get the vector (0, 1/2)^T. We can describe in general the projection of one vector u onto another, v by

where the dot represents the dot product. Since an inner product generalizes the idea of a dot product, then we have the equivalent formulation for any general inner product space:

where <v₁,v₂> represents the inner product.

This projection is indeed a projection, observe:

by definition, then

This projection is linear:

Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems:

• QR decomposition (see Householder transformation and Gram-Schmidt decomposition);
• Singular value decomposition
• Reduction to Hessenberg form (the first step in many eigenvalue algorithms).