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Extra info for Absorption and Space Reddening in the Galaxy as Shown by the Colors of Globular Clusters

Example text

We refer to the book [72]. The solution behavior of quadratic equations manifest themselves in a completely different way than for complex numbers. 11. (i) The reciprocally conjugate quaternions x = x0 + x and x = x0 − x satisfy the quadratic equation with real coefficients ξ 2 − 2x0 ξ + |x|2 = 0. (ii) A quaternion is a vector different from zero if and only if x2 is real and negative. (iii) A quaternion different from zero is real if and only if x2 is real and positive. (iv) The solution set of a quadratic equation in H with real coefficients consists either of one element, two elements or a two-dimensional sphere.

A vector y will be reflected through ρy about the plane spanned by y and e0 . Hence we cannot describe by means of ρy all the movements of H. An orthogonal matrix of fourth order can in general have as first case 4 real eigenvalues, which must all equal 1 in view of the length invariance; thus we obtain the identity. As second case two real eigenvalues can arise, and two complex conjugate ones. The former must once again equal 1, the corresponding eigenvectors determine an invariant plane, and those corresponding to the complex eigenvalues determine also a plane, in which one turns by a suitable angle.

An alternating application of both products is possible only with limitations, since in any case the vector product has to be calculated first. Here we can remark that multiple products are always to be expected with very specific properties. Instead the use of quaternion multiplication is completely without problems. The latter is computable without 2. Quaternions 43 problems and in addition associative, a fact that in the investigation of multiple products will play an important role. Let x, y, z be vectors.

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Absorption and Space Reddening in the Galaxy as Shown by the Colors of Globular Clusters by Stebbins J.


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