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- Find all monic irreducible polynomials of degree 2 over Z3.a.) Compute the characteristic polynomial of A. b.)Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.)b) Consider an N x N matrix A with N orthonormal eigenvectors xi such that Axi = λi xi , where the λi is the eigenvalue corrosponding to eigenvector xi . It can be shown that such a matrix A has an expansion of the form : A = (see image) i) Show that if the eigenvalues are real then A , as defined through the above expansion , is Hermitian ii) Using the result for A show that the N x N identify matrix can be witten as : I = (see image) iii) In proving this result for the identify matrix you have used the fact that the vectors {xi} are eigenvectors of a matrix A . Is this essential or is there a milder requirement possible? If so , what is it ? Justify your answers .
- b) Consider an N x N matrix A with N orthonormal eigenvectors xi such that Axi = λi xi , where the λi is the eigenvalue corrosponding to eigenvector xi . It can be shown that such a matrix A has an expansion of the form : (see image) i) Show that if the eigenvalues are real then A , as defined through the above expansion , is Hermitianb) Consider an N x N matrix A with N orthonormal eigenvectors xi such that Axi = λi xi , where the λi is the eigenvalue corrosponding to eigenvector xi . It can be shown that such a matrix A has an expansion of the form : (see image) ii) Using the result for A show that the N x N identify matrix can be written as : i = (see image)a) Without evaluating eigenvalues, determine the sum of eigenvalues of A.b) Without evaluating eigenvalues, determine the product of eigenvalues of A.c) Write the characteristic polynomial.
- a) Without evaluating eigenvalues, determine the sum of eigenvalues of A.b)Without evaluating eigenvalues, determine the product of eigenvalues of A.c) Write the characteristic polynomial.suppose --->x is an eigenvector of Acorresponding to an eigenvalue /|. (a) SHOW --->x is an eigenvector of 5I - 3A + A^2.(b) STATE its corresponding eigenvalue the arrow is over xLet G be a bipartite graph with adjacency matrix A. (a) Show that A is not primitive. (b) Show that if A is an eigenvalue of A, so is -λ.
- Working directly from definition, prove that if zn and wn are sequences of complex numbers withlimn→∞zn = 4 + 3i, and limn→∞wn = 4 − 3i,then limn→∞zn · wn = 25. (You may use the fact that convergent complex sequences are bounded.)Prove that (1,1) is an element of largest order in Zn1 + Zn2. State the general case. Needs Complete solution with 100 % accuracy.For all 3 of the given 2`-periodical functionsdefined on the interval of the length 2`, find1) expansion in Fourier series in real form;2) the function of the sum of the Fourier series for each value of xthat belongs to the interval;3) graph of the sum of the Fourier series.