Hello, does anyone know the solution of the below exercise 17? 17. Frame invaria

Question

Hello, does anyone know the solution of the below exercise 17?

17. Frame invariance and reciprocal screw systems An operator is said to be frame invariant if it does not depend on the choice of coordinate frame used to carry out the calculations Operations which are frame invariant can be computed relative to any coordinate frame, which can simplify calculations (a) Show that the reciprocal product between two screws is frame invariant. (b) Show that the inner product between two twists is not frame invariant. (c) Calculate a basis for the system of screws reciprocal to a zero- pitch screw through a point q. Give a geometric interpretation for the screws which form your basis. (Hint: perform your calculations relative to a specially chosen frame.) (d) Calculate a basis for the system of screws reciprocal to an infinite pitch screw. Give a geometric interpretation for the screws which form your basis (e) Using reciprocal screws, show that three parallel, coplanar, zero-pitch screws are dependent. That is, exhibit a system of four independent screws which are reciprocal to each of the coplanar ScrewS

Explanation / Answer

a] If Transpose of the Complex Conjugate of a given matrix is the initial matrix itself, then the given matrix is Hermitian.

Here,

so, its complex conjugate will be:

and its transpose will be:

hence the given matrix is Hermitian.

b]

so,

and its determinant should be zero in order to satisfy the eigen value equation. And so:

=>

=>

=>

=>

therefore, the eigen values are: and .

Please upvote if the solution was helpful :)

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