4. Representing Vector w.r.t. Another Basis (20 points, each question 5 points) - Any orthonormal matrix U=u1u2un defines a new basis of Rn. - For any vector xRn can be represented as a linear combination of u1,,un, with coefficient A^1,,A^n : x=^1u1++^nun=Ux^ - Indeed, such x^ uniquely exists x=Ux^Ux=x^ In other words, the vector x^=Ux can serve as another representation of the vector x w.r.t the basis defined by U. Let A be a square matrix, so that we can do eigendecomposition: A=UUT Based on the above information. (1) Show that "left-multiplying matrix A can be viewed as left-multiplying a diagonal matrix w.r.t the basis of the eigenvectors," that is, for z=Ax, show z^=Uz=x^=1x^12x^2nx^n (2) Suppose q= AAAx, show that q^=Uq=3x^=13x^123x^2n3x^n (3) Show that xTAx=i=1nix^i2 (4) Using (3)'s conclusion, show that If all i>0, then the matrix A is sositivedefinite