Bivectors and Waves in Mechanics and Optics by Ph. Boulanger, M. Hayes (auth.)

By Ph. Boulanger, M. Hayes (auth.)

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A. Thus A and Bare linearly independent. 1 :f. A. 2 (Synge) If A and B are eigenbivectors of Q corresponding to different eigenvalues, then A and B cannot both be isotropic. Proof Indeed, A and B are orthogonal and linearly independent. 2) which contradicts the statement that A and B are linearly independent. 2 Isotropic eigenbivectors Here we consider the possibility that the complex symmetric matrix Q has isotropic eigenbivectors. 2 ISOTROPIC EIGENBIVECTORS 43 an isotropic eigenbivector is that the matrix have a double or triple eigenvalue.

Thus A and Bare linearly independent. 1 :f. A. 2 (Synge) If A and B are eigenbivectors of Q corresponding to different eigenvalues, then A and B cannot both be isotropic. Proof Indeed, A and B are orthogonal and linearly independent. 2) which contradicts the statement that A and B are linearly independent. 2 Isotropic eigenbivectors Here we consider the possibility that the complex symmetric matrix Q has isotropic eigenbivectors. 2 ISOTROPIC EIGENBIVECTORS 43 an isotropic eigenbivector is that the matrix have a double or triple eigenvalue.

B) If Cis isotropic and A, Bare not isotropic, then C* may be isotropic, and A* orB* (but not both) may also be isotropic. 11. 7) and the bivector A* (B*) is isotropic when B· C = 0 (c- A = 0). 7 1. Let A = i + ij, B = j + ik. Choose C such that A* and B* are isotropic. Determine A*, B*, C*. 2. Let A= i- ij- k, B = i + ij + k, C = i + ij. Find A*, B*, C*. How many isotropic bivectors are contained in the triad A, B, C and A*, B*, C*? 12 Bivector decompositions Given an arbitrary bivector C (not with a real direction), it may always be written as a linear combination of two linearly independent bivectors in its plane.

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