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Mathematics for Physics

264 Projection Map

A linear transformation that maps every vector into a fixed subspace and leaves vectors in that subspace unchanged that is used to extract the component along a chosen subspace.

Note: Also called projection. Also called projection operator. Also called orthogonal projection when the leftover part is perpendicular to the subspace.

definition [d] (Orthogonal Projection) From Axler: suppose \(U\) is a finite-dimensional subspace of \(V\). The orthogonal projection of \(V\) onto \(U\) is the operator \(P_{U} \in \mathcal{L}(V)\) defined as follows. For each \(v \in V\), write \(v = u + w\), where \(u \in U\) and \(w \in U^{\perp}\). Then let

  • \(P_{U} v = u\) .

where

  • \(V\) is an inner product space.
  • \(U\) is a finite-dimensional subspace of \(V\).
  • \(U^{\perp}\) is the orthogonal complement of \(U\).
  • \(P_{U}\) is the orthogonal projection onto \(U\).

definition [d] (Projection) From Hubbard and Hubbard: a square \(n \times n\) matrix \(P\) such that

  • \(P^{2} = P\)

is called a projection.

where

  • \(P\) is an \(n \times n\) matrix.
  • \(P^{2}\) means the matrix product \(PP\).

definition [d] (Projection Operator) From Griffel: let \(S\) be a subspace of the Hilbert space \(H\). The operator \(P: H \rightarrow S\) defined by \(Px = y\), where \(y\) is the projection of \(x\) onto \(S\), is called the projection operator onto \(S\). Here each \(x \in H\) has a unique splitting \(x = y + z\) with \(y \in S\) and \(z\) orthogonal to \(S\).

where

  • \(H\) is a Hilbert space.
  • \(S\) is a subspace of \(H\).
  • \(y\) is the projection of \(x\) onto \(S\).
  • \(P\) is the projection operator onto \(S\).

264.1 Elementary Example

264.1.1 Simple

Projection onto the \(x\)-axis in the plane maps \((x,y)\) to \((x,0)\) and satisfies \(P^{2}=P\).

\[ P\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ 0 \end{pmatrix} \]

\[ P = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\quad P^{2} = P \]

where

  • \(P\) is the projection map onto the \(x\)-axis.
  • \(x\) and \(y\) are real coordinates.

264.1.2 General

Orthogonal projection onto the \(xy\)-plane in three dimensions keeps the first two coordinates and drops the third.

\[ P_{U}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} \]

\[ U = \left\{ \begin{pmatrix} a \\ b \\ 0 \end{pmatrix} : a,b \in \mathbb{R} \right\} \]

\[ v = u + w,\quad P_{U} v = u \]

where

  • \(U\) is the \(xy\)-plane subspace.
  • \(u\) is the part of \(v\) in \(U\).
  • \(w\) is the leftover part orthogonal to \(U\).

264.2 References

  1. Axler, S. Linear Algebra Done Right, 4th ed. Springer, 2024. — Def. 6.55: orthogonal projection \(P_{U}\) with \(v = u + w\), \(u \in U\), \(w \in U^{\perp}\), and \(P_{U}v = u\).
  2. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Matrix Editions, 2015. — Ex. 2.18: a square matrix \(P\) with \(P^{2} = P\) is a projection.
  3. Griffel, D. H. Applied Functional Analysis. Ellis Horwood, 1981/1985. — Def. 9.34: projection operator \(P: H \rightarrow S\) with \(Px = y\) the projection of \(x\) onto \(S\).