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The orthogonal projection is the -component in
Definition
Let be a subspace of a finite-dimensional inner product space . The orthogonal-decomposition theorem (For a subspace of a finite-dimensional inner product space, ) gives
associates to every unique vectors and with . The orthogonal projection onto is the function
Equivalently, is the unique vector of such that .
Depends on
Used by
- An invariant subspace need not reduce an operator Counterexample
- The Hilbert orthogonal projection onto a closed subspace Definition
- Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ Proposition
- A normal endomorphism is a sum of its eigenvalues times pairwise orthogonal projections, and each spectral projection is a polynomial in the endomorphism Theorem
- An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint Theorem
- The eigenvalues of the orthogonal compression of a self-adjoint endomorphism to a hyperplane interlace those of the original endomorphism Theorem
- The orthogonal projection is the unique nearest point in the subspace Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., definition 6.55 (standard reference, not scraped)
- Sergei Treil, Linear Algebra Done Wrong, Definition 5.3.1 (standard reference, not scraped)