The orthogonal projection $P_{W}v$ is the $W$-component in $V=W\oplus W^{\perp }$

Depends on

• For a subspace $W$ of a finite-dimensional inner product space, $V=W\oplus W^\perp$

Used by

• An invariant subspace need not reduce an operator Counterexample
• Fixed-design least squares as orthogonal projection Definition
• Polyhedral selection event and contrast interval Definition
• Projected-unitary encoding and singular vectors Definition
• Series least squares with a spline basis Definition
• The Hilbert orthogonal projection onto a closed subspace Definition
• Threshold-band singular-value approximation Definition
• Finite transitive G-sets recover the stabilizer-induction classification Example
• Leverage is determined by the design Example
• Exclusions for positive definite diagrams: trees, valency, labels, chains and arms Lemma
• Finite Coxeter orbit polytopes, face isometries and their cocycle Lemma
• Finite-dimensional orthogonal projections are linear and contractive Lemma
• Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas Lemma
• Nested contrast models and projections in a balanced block design Lemma
• Operator perturbation bound for QSVT Lemma
• Orthogonal error restrictions determine syndrome projections Lemma
• Positive Marginals Imply Joint Support Lemma
• Projected coordinate vectors give a deterministic orthonormal basis Lemma
• Projection onto a finite-dimensional subspace of an arbitrary real inner-product space Lemma
• PUE singular-space invariant decomposition Lemma
• Singular-value transform is independent of SVD choices Lemma
• The standard Wishart inverse quadratic form has an exact Fisher law Lemma
• The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order Lemma
• 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
• BIC fitted-value risk bound Theorem
• Exact squared-error risk of an isotropic projection fit Theorem
• Exact t and partial F inference in full-rank Gaussian regression Theorem
• Expected empirical training risk for spline series least squares Theorem
• Least-squares fitted vectors and residuals are unique Theorem
• Orthogonal projections split a standard Gaussian into independent chi-square sums Theorem
• Phase estimation on a superposition of eigenspaces Theorem
• QR decomposition gives exact t and F pivots in the fixed Gaussian linear model Theorem
• QSVT robustness, approximation error, and resources Theorem
• Ray-to-projector is well-defined and bijective Theorem
• Sobolev approximation by local spline quasi-interpolants Theorem
• Sparse constrained least-squares risk bound Theorem
• Sub-Gaussian projection risk bounds Theorem
• The eigenvalues of the orthogonal compression of a self-adjoint endomorphism to a hyperplane interlace those of the original endomorphism Theorem

…and 2 more results.

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)