7. Consider the symmetric matrix A = [4556] .

(a) Find an invertible matrix M and a diagonal matrix D such that M^T AM = D.

(b) Consider the quadratic form Q(x) = x^T Ax where x = [x1x2] R².

Find an invertible matrix P such that y = Px and the quadratic form has no cross

terms, i.e., Q(y) = Q(y₁,y₂) = ay₁² + by₂² for some a,b R.

7. Consider the symmetric matrix A = CT CT (a) Find an invertible matrix M and a diagonal matrix D such that MAM = D. (b) Consider the quadratic form Q(x) = x Ax where x = E IR 2 . Find an invertible matrix P such that y = Px and the quadratic form has no cross terms, i.e., Q(y) = Q(y1, y2) = ayr + by? for some a, b E R