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For a linear map between finite-dimensional inner-product spaces, minimises if and only if , equivalently ; minimisers exist and any two differ by an element of
Statement
Let be a linear map between finite-dimensional inner product spaces and let . A vector minimises if and only if
equivalently . Minimisers exist, and if is one minimiser, then the full set of minimisers is .
Facts & Assumptions
Given: A finite-dimensional map and .
Orthogonal projection onto a finite-dimensional subspace is its unique nearest point (The orthogonal projection is the unique nearest point in the subspace).
The adjoint is defined by (The adjoint is characterised by ).
The identity holds in finite dimension ( and in finite dimension).
Proof
The subspace has the unique nearest point to by [L1]. Choose with . Thus a minimiser exists.
A vector is a minimiser exactly when , which by orthogonal projection is exactly when . By [L3], this is exactly .
Linearity turns the last equation into , equivalently .
If and are minimisers, uniqueness of the nearest image point gives , so . Conversely, adding any element of leaves the image and residual unchanged. Hence the minimisers are exactly .
Depends on
Used by
- Flexible GMRES with variable right preconditioning Algorithm
- The MINRES iterate from the Lanczos tridiagonal least-squares problem Definition
- The minimum-norm indicator-response least-squares fit exists uniquely Lemma
- A nonbreakdown LSQR iterate minimizes residual on its Krylov space Theorem
- Arnoldi reduces GMRES to a least-squares problem for the small Hessenberg matrix Theorem
- Binary logistic derivatives and the exact Newton/IRLS update Theorem
- Embedding span(A,b) gives the sketch-and-solve residual factor Theorem
- Every least-squares solution has the form A^+b+(I-A^+A)z, and the same affine family specializes to exact solutions when b inimA Theorem
- Finite LLE affine least-squares weights have a unique minimum-norm minimizer Theorem
- Flexible GMRES minimizes over the actual preconditioned-vector span Theorem
- For every right-hand side b, A^+b is the unique least-squares solution of minimum Euclidean norm Theorem
- For full-column-rank A, the normal equations square the spectral condition number Theorem
- IRLS equals the Fisher-scoring step when information is invertible Theorem
- Kernel ridge regression equals the Gaussian-process posterior mean Theorem
- Least-squares fitted vectors and residuals are unique Theorem
- Lloyd iteration is objective-decreasing and terminates Theorem
- QR decomposition gives exact t and F pivots in the fixed Gaussian linear model Theorem
- Reduced QR over the reals solves full-column-rank least squares without squaring the condition number Theorem
- The 3DVAR quadratic has the Gaussian MAP and Kalman analysis update Theorem
- The Frobenius least-squares objective has gradient A^*(Ax-b) and Hessian A^*A in the vector variable Theorem
- The full ordinary least-squares solution set and the unique ridge solution Theorem
- The sample mean uniquely minimizes squared error on a nonempty leaf Theorem
- Zero-initialized gradient descent selects minimum-norm least squares Theorem
Dependency tree · two levels
10 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
- Sergei Treil, Linear Algebra Done Wrong, §5.4.1 (standard reference, not scraped)