i need step by step detail and hand written solution

Problem 1 [10 points] Let A be a real n x n symmetric matrix. We say that A is positive semidefinite if and only if x Ax 2 0, for every vector x E R". Prove the following statements, which were claimed in the video lectures. (i) Prove that a real symmetric matrix has real eigenvalues (the point being that a general matrix may have complex eigenvalues). (ii) Prove that a real symmetric matrix has orthogonal eigenvectors. Then, prove that the eigen-decomposition of a real symmetric matrix A is A = QAQT. (iii) Prove that a positive semidefinite matrix has non-negative eigenvalues. (iv) Let A be a positive semidefinite matrix. Define matrix D so that D; = A; + A; -2A;;. Show that there exist n vectors {v] , . ..,V,} in R" so that D,; = liv, - v, ll