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Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
Definition
A linear map (Linear map between vector spaces over the same field) between inner product spaces (Real and complex inner product spaces, with the inner product linear in the first argument) is a linear isometry if
for every , where the norm is the induced norm The norm induced by a real or complex inner product. An invertible linear isometry from a real finite-dimensional inner product space to itself is an orthogonal operator; over it is a unitary operator.
Equivalently, once the finite-dimensional characterisation is proved, orthogonal and unitary operators are the endomorphisms satisfying .
Depends on
Used by
- Lebesgue measure on ℝⁿ is invariant under every orthogonal linear map Corollary
- Left and right regular unitary representations of an LCH group Definition
- Matching C¹ pieces across a hyperplane have no jump derivative Example
- Newtonian potential of radial compact data Example
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- Pseudoinversion is involutive, commutes with adjoints, and is equivariant under unitary left and right factors Proposition
- Every endomorphism has a polar decomposition T = SU with U non-negative and S an isometry on the orthogonal complement of ker T, and S is unique exactly when T is invertible Theorem
- For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent Theorem
- For full-column-rank A, the normal equations square the spectral condition number Theorem
- Householder reflectors and Givens transformations are unitary and can annihilate prescribed entries Theorem
- Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison Theorem
Dependency tree · two levels
8 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., definitions 7.44 and 7.51 (standard reference, not scraped)