Revision #618 → #1449 · back to history
addedOrthogonal matrix6939a7165364
addedTranspose equals inverse characterization65c2f0e7811d
addedBasic properties (invertible, unitary, normal, isometry)c009e1eee213
addedOrthogonal group O(n)5d679a21b97f
addedSpecial orthogonal group SO(n)8885c8aa3c95
addedReal specialization of unitary, always normaled51838f9536
addedOrthogonal matrices preserve the dot productd1cf9a3e3ce8
addedConverse: orthogonal matrices imply orthogonal transformations0dabb19094df
addedIdentity transformation5dbca2ea614c
addedRotation about the origin64a75be3e122
addedReflection across x-axis17284f0d3f32
addedPermutation of coordinate axes28c5f19db711
added1×1 orthogonal matrices5dbd68862e7a
added2×2 orthogonality equationsc6f1f7e80085
addedRotation and reflection cases of 2×21c2902656d87
added90° reflection is a permutation matrixf92a7f3c4723
addedIdentity is a permutation matrix675c01ed8ab5
addedReflection matrix is symmetric65655268b70f
addedProducts of rotation/reflection matrices31e154ba14fc
addedClassification as rotational or notc74c2b5286fa
addedInversion and rotoinversion5c34ab670fa6
addedHigher-dimensional rotations need several anglesd3d3102d5b3c
addedTranspositionf44c61bea8bd
addedPermutation matrix as product of transpositions877312d9f1a8
addedHouseholder reflection4a3d38e6ed44
addedOrthogonal matrix as product of Householder reflectionsa6d5a57a99a9
addedGivens rotation5e16a173bf42
addedRotation matrix as product of Givens rotations5ec7434978d2
addedJacobi rotation776fbf9f2286
addedOrthogonal iff columns form orthonormal basis0bb062246bf4
addedDeterminant is ±19d5a76f0ffed
addedDeterminant ±1 does not imply orthogonality8853423c5368
addedPermutation determinant equals signature5fe680594c93
addedDiagonalizable over ℂ with eigenvalues of modulus 1c1c3966b7b72
addedOrthogonal matrices form a compact Lie group55272301ea59
addedSO(n) is a normal subgroup of index 2a970fbe93775
addedO(n) is a subgroup of O(n+1)1fd6db645867
addedOrthogonal group is a reflection group3c18b7eb8621
addedSO(n) is a subgroup of SO(n+1)46dec953e711
addedPermutation matrices form the symmetric group20fc54fba6cb
addedBlock-diagonal decomposition of orthogonal matricesda970e44247d
addedCanonical form of any orthogonal matrix23dd53bfd4f3
addedEigenvalues on unit circle; real eigenvalue when n oddb1b2511f215f
addedLie algebra consists of skew-symmetric matrices9f3aa931b519
addedExponential of skew-symmetric is orthogonalb67748a1ca98
addedAngular velocity as Lie algebra element7087e52d696e
addedExponential gives axis-angle rotation matrix9a577e6ea3d1
addedCondition number is 113f9b36d66b3
addedQR decomposition7784014d1998
addedSingular value decomposition83339aa244d2
addedSpectral decomposition of symmetric matrix2689ad45469b
addedPolar decomposition8df9d2d06214
addedOverdetermined system via QR6053a1368f65
addedLinear least squares problem4c982e30d35d
addedUnderdetermined system via SVD pseudoinverse92fc8b51cd4e
addedPolar decomposition gives nearest orthogonal matrixf8a20238d6f9
addedAveraging algorithm convergence9b816c6bff29
addedUniform via Haar measure3467fc41d312
addedQR of normal entries yields uniform orthogonal matrices5b1503ae298f
addedSubgroup algorithm for random orthogonal matrices4c81b5c39f54
addedNearest orthogonal matrix via SVD6aebb9811295
addedQuadratically convergent recurrence2fa96659a89b
addedStability condition for iterationsc25689aff6f6
addedSO(n) is not simply connected5344864145ff
addedSpin(n) is the universal cover of SO(n)5bf1d3dd0fcb
addedSpin(3) = SU(2)2726d6c6a469
addedNon-square: the two conditions differ1e8788d1b355
addedOrthogonal k-frames and Stiefel manifold86465c1cd4a9