Problem 7 (15 points). Suppose A E Rnxn. Let 1, 12, ..., An E C be all eigenvalues of A (they may or may not be distinct). Also, let vi (with | |vill2 = 1) for ie {1, ...,n} be an eigenvector associated with di. a. (2 points) Suppose Al = A. Prove Vie {1, .., n} : MER. b. (2 points) Suppose Al = A. Prove that there exists a set of eigenvectors of A which is an orthogonal basis for Rn. c. (2 points) Suppose {v1, ..., Un} is an orthogonal basis for R". Prove AT = A. d. (3 points) Suppose AT = A and Amin = mini di, Amaz = maxi Mi. Prove VER" : AminIII AT LAmar ITI e. (3 points) Prove that 1, 12, ..., An are also eigenvalues of AT. 3 f. (3 points) Suppose A is a positive definite matrix, that is, AT = A and for any r E R" : I Ar > 0. Prove Vie {1, ..., n} : di > 0