Eutactic stars are useful largely because of their relationship with the geometry of polytopes and groups of orthogonal transformations.
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This moves as much of the variance as possible ( using an orthogonal transformation ) into the first few dimensions.
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In 1972, she defended her doctoral thesis, " The Use of Orthogonal Transformations in Solving Problems of Algebra ."
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The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction.
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PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest " variance " ( as defined above ).
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In finite-dimensional spaces, the matrix representation ( with respect to an orthonormal basis ) of an orthogonal transformation is an orthogonal matrix.
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:A star is eutactic if it is transformed to itself by some irreducible group of orthogonal transformations that acts transitively on pairs of opposite vectors.
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Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them.
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This procedure has some similarities to the procedure for working with homogeneous coordinates in projective geometry and in this case allows the modeling of Euclidean transformations as orthogonal transformations.
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However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.
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