1
answer
0
watching
126
views
13 Nov 2019
Can anyone help with these two linear algebra problems please?
2) (Adapted from an older edition of Peter Lax's book:) We define the cyclic shift mapping S, acting on vectors a e Cn, by S(x1,T2, . . . ,xn) = (xn , X 1 , ,Xn-1). Prove that S preserves the norm of vectors, and is therefore unitary (the name for orthogonal matrices with complex entries.) Determine the eigenvalues and eigenvectors of S, and verify that the eigen- values have absolute value 1 and that the eigenvectors are indeed orthog- onal. . Verify the following surprising fact: the expansion of a vector v in terms of the eigenvectors of S is the discrete Fourier transform of v 3) Think of Cn as the space of discrete periodic functions; i.e. identify an+1 with a1 and zo with zn. Consider the linear transformations corresponding to the discrete first (forward) and second derivatives, Find their eigenvalues and eigenvectors. (Hint: Do they commute with the transformation S of the second exercise?)
Can anyone help with these two linear algebra problems please?
2) (Adapted from an older edition of Peter Lax's book:) We define the cyclic shift mapping S, acting on vectors a e Cn, by S(x1,T2, . . . ,xn) = (xn , X 1 , ,Xn-1). Prove that S preserves the norm of vectors, and is therefore unitary (the name for orthogonal matrices with complex entries.) Determine the eigenvalues and eigenvectors of S, and verify that the eigen- values have absolute value 1 and that the eigenvectors are indeed orthog- onal. . Verify the following surprising fact: the expansion of a vector v in terms of the eigenvectors of S is the discrete Fourier transform of v 3) Think of Cn as the space of discrete periodic functions; i.e. identify an+1 with a1 and zo with zn. Consider the linear transformations corresponding to the discrete first (forward) and second derivatives, Find their eigenvalues and eigenvectors. (Hint: Do they commute with the transformation S of the second exercise?)
Lelia LubowitzLv2
21 Feb 2019