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2021
In the last lecture, we introduced the description of rigid body motion in the three-dimensional world, including the rotation matrix, rotation vector, Euler angle, quaternion, and so on. We focused on the representation of rotation, but in SLAM, we have to estimate and optimize them in addition to the representation.
Xiang Gao, Tao Zhang
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In the last lecture, we introduced the description of rigid body motion in the three-dimensional world, including the rotation matrix, rotation vector, Euler angle, quaternion, and so on. We focused on the representation of rotation, but in SLAM, we have to estimate and optimize them in addition to the representation.
Xiang Gao, Tao Zhang
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2004
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
William Fulton, Joe Harris
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In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
William Fulton, Joe Harris
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2012
We have studied linear transformation on \({\mathbb{R}}^{n}\) using the traditional matrix formalism in Chap. 7 and more generally in Chaps. 8–10, using the machinery of geometric algebra. This chapter explains the bivector interpretation of a general linear operator and offers a new proof of the Cayley–Hamilton theorem based upon this interpretation.
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We have studied linear transformation on \({\mathbb{R}}^{n}\) using the traditional matrix formalism in Chap. 7 and more generally in Chaps. 8–10, using the machinery of geometric algebra. This chapter explains the bivector interpretation of a general linear operator and offers a new proof of the Cayley–Hamilton theorem based upon this interpretation.
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Differential Geometry, Lie Groups, and Symmetric Spaces
, 1978S. Helgason
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Introduction to Lie Algebras and Representation Theory
, 1973J. Humphreys
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Lie superalgebras and Lie supergroups. II
1991These two papers represent the transcript of a seminar giving a survey of the subject. After the definitions, including the series of simple algebras \(A\), \(B\), \(C\), \(D\) defined as super-algebras, Grassmann-hulls are introduced, as block matrices over the tensor product of a Lie super algebra and Grassmann algebra.
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1976
As we pointed out in 6.2, there are exactly two simple real Lie algebras of dimension 3. These are: the algebra \( {{\mathfrak{g}}_{1}} = \mathfrak{s}\mathfrak{l}\left( {2,R} \right) \) of real matrices of the second order with zero trace and the algebra \( {{\mathfrak{g}}_{2}} = \mathfrak{s}\mathfrak{o} = \left( {3,R} \right) \) of real skew-symmetric
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As we pointed out in 6.2, there are exactly two simple real Lie algebras of dimension 3. These are: the algebra \( {{\mathfrak{g}}_{1}} = \mathfrak{s}\mathfrak{l}\left( {2,R} \right) \) of real matrices of the second order with zero trace and the algebra \( {{\mathfrak{g}}_{2}} = \mathfrak{s}\mathfrak{o} = \left( {3,R} \right) \) of real skew-symmetric
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