Linear and multilinear algebra; matrix theory Definition
Orthogonal vectors
Builds on Inner product
Definition
Let be an inner product space over or . Two vectors are orthogonal, written , when , where is the inner product.
Remarks
The relation is symmetric, because . The zero vector is orthogonal to every vector, and it is the only vector orthogonal to itself. In with the dot product, and are orthogonal.
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Vector space definition
- Inner product definition
- Orthogonal vectors definition
Where this leads
80 pages build on this one.
- Reflection
- Parseval's identity
- Spectral theorem
- Orthogonal complement
- Coordinates in an orthonormal basis
- Orthonormal family
- Pythagorean theorem in an inner product space
- Least squares
- Matrix of the adjoint is the conjugate transpose
- Spectral theorem for Hermitian matrices
- Dimension of the orthogonal complement
- QR decomposition
- Orthonormal basis
- Eigenspaces of a self-adjoint operator are orthogonal
- An orthonormal family is linearly independent
- Orthogonal projection is the closest point
- Principal axis theorem
- Orthogonal complement of the orthogonal complement
- Orthogonal projection
- Bessel's inequality
- A space is the direct sum of a subspace and its orthogonal complement
- Cartan–Dieudonné theorem
- Gram–Schmidt process
- Gram–Schmidt vectors are orthogonal
- Existence of an orthonormal basis
- Non-negative least squares
- Derivation of the conjugate gradient method
- Conjugate residual method
- Local regression
- Hilbert–Schmidt expansion theorem
- Minimal residual method
- Toeplitz operator
- Partial isometry
- Céa's lemma
- Bilinear series of a symmetric kernel
- Ordinary least squares
- QR algorithm
- Smoothing spline
- Lax–Milgram theorem
- Hadamard's inequality in Fredholm theory
- Response surface methodology
- Nonlinear regression
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Generalized minimal residual method
- Minimum total potential energy principle
- Hadamard's inequality
- Arnoldi iteration
- Gauss–Newton algorithm
- Hilbert space adjoint
- Lidskii's trace formula for integral operators
- Normal coordinates
- Hadamard's maximal determinant problem
- Spectrum of the Laplace–Beltrami operator
- Hilbert projection theorem
- Ridge regression
- L-curve
- Biconjugate gradient stabilized method
- Galerkin method
- Bernstein's theorem on absolutely convergent Fourier series
- Operator monotone function
- Empirical risk minimization
- Isotonic regression
- Projection body
- Mercer's theorem
- Principal component analysis
- Ladyzhenskaya–Babuška–Brezzi condition
- Levenberg–Marquardt algorithm
- Schur product theorem
- Fundamental theorem of space curves
- Riesz representation theorem
- Box–Jenkins method
- Singular values of a compact operator
- Lasso
- Householder transformation
- Numerical methods for linear least squares
- Schmidt's singular value decomposition of a kernel
- Three-term recurrence for orthogonal polynomials
- Bareiss algorithm
- Frenet–Serret formulas
Further reading
- Web https://en.wikipedia.org/wiki/Orthogonality_(mathematics), Euclidean vector spaces
About this page
| Tag | 0TS7VZJBWD |
|---|---|
| URI | m2://0TS7VZJBWD |
| Kind | definition |
| Writing | W0 stub |
| MSC | 15A63, 46C05 |
| Defines | Orthogonal vectors |
| Uses | Inner product |
Markdown · JSON · Edit on GitHub
What this page needs for W1
- an inventory row: the row of the inventory that the node accounts for (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.