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Unformatted text preview: Homework 2 Classical Mechanics (Phys701) Due: September 19, 2011 Problem 4-2 Prove the following properties of the transposed and adjoined matrices ( A B ) T = B T A T , ( A B ) = B A Solution: [ ( A B ) T ] ij = A jl B li = ( B T ) il ( A T ) lj = ( A T B T ) ij [ ( A B ) ] ij = A jl B li = ( B ) il ( A ) lj = ( A B ) ij Problem 4-3 Show that the trace of the matrix is invariant under any similarity transformation. Show also that antisymmetric property of a matrix is preserved under orthogonal similarity transformation. Solution: If the transformation is given by a matrix S then A A = S 1 AS Hence for the trace Tr A = ( S 1 ) im A mn S ni = S ni ( S 1 ) im A mn = ( SS 1 ) nm A mn = nm A mn = A nn = Tr A When the transformation is a rotation S = R , S 1 = R T , and A ij = A ji one gets A ji = ( R T ) jm A mn R ni = ( R T ) jm A nm R ni = ( R T ) in A nm R mj = A ij 1 Problem 4-11 Verify the relationship det( B ) = ( 1) n det B for the determinant of an n n matrix B . Solution: We write B = ( E ) B . Then det( B ) = det( E )det B . Matrix E is diagonal with 1 on the diagonal. Its determinant is a product of n factors ( 1). Thus we get det( B ) = ( 1) n det B . 1 Problem 4-12 In a set of axes where z axis is the axis of a finite rotation, the rotation matrix is given by equation (4.43) R = cos sin sin cos 1 Derive the rotation formula (4.62) r = r cos + n ( n r )(1 cos ) + [ r...
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