Consider a N dimensional vector expressed in the identity basis. (a) Express the vector in an orthonormal basis F, where F is a N * N matrix. (Hint : See class notes on how we express a signal in different basis.) (b) Let's call the above vector . Now create a matrix B such that Bw scales the th element of by a scalar bi. What should be the matrix B? (c) Let's denote the vector Bw as vector z. Now convert vector z back into the original identity basis. (d) Now write all the above operations on vector in one equation in terms of F and B. (e) Write the Eigen-decomposition equation of a matrix A, where S contains the eigenvectors of A and A is the diagonal matrix containing the eigenvalues. (f) Given the above exercise you have done, explain in plain English what Eigen-decomposition does to a vector (in other words, what happens when matrix A is multiplied to vector x)? Consider a N dimensional vector expressed in the identity basis. (a) Express the vector in an orthonormal basis F, where F is a N * N matrix. (Hint : See class notes on how we express a signal in different basis.) (b) Let's call the above vector . Now create a matrix B such that Bw scales the th element of by a scalar bi. What should be the matrix B? (c) Let's denote the vector Bw as vector z. Now convert vector z back into the original identity basis. (d) Now write all the above operations on vector in one equation in terms of F and B. (e) Write the Eigen-decomposition equation of a matrix A, where S contains the eigenvectors of A and A is the diagonal matrix containing the eigenvalues. (f) Given the above exercise you have done, explain in plain English what Eigen-decomposition does to a vector (in other words, what happens when matrix A is multiplied to vector x)