posdef-hw

Definiteness Name: NetID: Problem 1. Let q ( x ) = ⟨ x , S x ⟩ where S = 10 3 3 2 Complete the square to write q ( x ) = λ 1 y 2 1 + λ 2 y 2 2 and determine the definiteness of S . Write each of y 1 and y 2 in terms of x 1 and x 2 .

Problem 2. Suppose that S 1 and S 2 are positive definite matrices. Show that S 1 + S 2 is also positive definite. Hint. Consider q ( x ) = ⟨ x , ( S 1 + S 2 ) x ⟩ . Problem 3. Suppose d 1 , d 2 , and d 3 are real numbers and consider the matrix S given by S = L   1 0 0 − 3 1 0 7 − 9 1   D   d 1 0 0 0 d 2 0 0 0 d 3   L ⊺   1 − 3 7 0 1 − 9 0 0 1   Let q ( x ) be the quadratic form q ( x ) = ⟨ x , S x ⟩ . ( a ) Prove that S is real-symmetric without calculating S . ( b ) It is possible to write q ( x ) = d 1 y 2 1 + d 2 y 2 2 + d 3 y 2 3 . Find y 1 , y 2 , and y 3 in terms of x 1 , x 2 , and x 3 . ( c ) Under what condition is S positive definite? Clearly explain your condition.