Abstract
Linear algebra is the language describing systems in finite (or countably infinite) dimensions, where dimension represents the number of variables at hand.
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Notes
- 1.
Formally \(I_{nn}\), since I is generally a two-index tensor.
- 2.
Ernst Mach (1838–1916).
- 3.
- 4.
- 5.
The complete set of frame dragging induced interactions is described by the Riemann tensor.
- 6.
The product of the eigenvalues equals the determinant of the matrix, as follows from, e.g., the Jordan decomposition theorem. The same theorem shows that the trace of a matrix given by the sum of the elements on the principle diagonal equals the sum of the eigenvalues.
- 7.
Also referred to as the Hermitian transpose or the conjugate transpose.
- 8.
The celestial sphere in the language of cosmology.
- 9.
Commonly referred to as SO(3), described by rotation matrices with determinant +1.
- 10.
Photons carry an additional degree of freedom in polarization.
- 11.
Wolfgang Pauli 1900–1958.
- 12.
The image space is comprised of multiples of one vector \(\left( \begin{array}{cc} \xi&\eta \end{array}\right) ^\dagger \).
- 13.
\(S^1\) is illustrative of a one-dimensional manifold which is compact and simply connected. It has nontrivial topology, since the winding number of a loop in \(S^1\) can take any value in \(\mathbb Z\). By homotopy, the topology of \(S^1\) is the same as that of the punctured disk \(0<|z|\le 1\).
- 14.
As a result, we say \(SU(2)\,\subset \, U(2)\cong SU(2)\,\times \,U(1)\).
- 15.
The symbols \(o^A\bar{\iota }^{A'}\) and \(\bar{\iota }^{A'}o^A\) are the same, i.e., there is no ordering between unprimed and primed indices. Only upon expansion into a matrix, a choice of ordering is made.
- 16.
A so-called Rindler space.
- 17.
It can be shown that flatness is preserved under analytic continuation, whereby the Lorentz metric \(ds^2=-dt^2+dx^2=(idt)^2+dx^2\) is trivially flat.
References
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Kerr, R.P., 1963, Phys. Rev. Lett., 11, 237.
van Putten, M.H.P.M., 2005, Nuov. Cim. B, 28, 597; van Putten, M.H.P.M., & Gupta, A.C., 2009, Mon. Not. R. Astron. Soc., 394, 2238.
Papapetrou A., 1951, Proc. R. Soc., 209, 248.
Feynman, R.P., 1963, Lectures on Physics, Vol. I (Addison-Wesley Publishing Co.), Ch. 20.
van Putten, M.H.P.M., 2017, NewA, 54, 115, arXiv:1609.07474.
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van Putten, M.H. (2017). Vectors and Linear Algebra. In: Introduction to Methods of Approximation in Physics and Astronomy. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-10-2932-5_3
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