Let X be a binomial random variable as in Example 33.5. Prove that E.X / D np

M303 Section 5.1 Notes- Introduction to “Eigenstuff”; Diagonalization 11-30-16  From here on, only consider square matrices ( )  If is × , then gives rise to liner map :ℝ → ℝ givn by = o Since source and target dimensions are identical, can compare inputs and outputs  Focus on vectors on which has particularly simple effect  Eigenvector of matrix - nonzero vector such that = , where ℝ o uniquely determined and called the eigenvalue of (for ); we call a -eigenvector o Eigenvalues of are scalars for which there exist -eigenvectors  may not have any eigenvalues or eigenvectors  E-vectors nonzero by definition ( = , would not be unique), but e-values can be 0  Ex. Let = [ −2], = (−1,1), and = (2,1). Is either or an e-vector of 1 0 o = [3 −2 ][ ] 1 0 1  = [−3 − 2 ] −1 − 0  = [−5 ] −1  ≠ → not e-vector o = [3 −2 2][ ] 1 0 1  = [6 − 2] 2 − 0  = [ ] 2  = 2 → is e-vector with e-value 2 1 6  Ex.