. (a) Let A be an n - n-symmetric matrix. Diagonalize A to show that x • Ax || x || ² is greater than or equal to the smallest eigenvalue of A for all x 0 in R". (b) Show that the quadratic form Q₁(x) = x Ax is coercive if and only if A is positive definite. (c) Conclude from (b) that if f(x) = a + b x + x• Ax is any quadratic function where a € R, be R" and A is an n × n-symmetric matrix, then f(x) is coercive if and only if A is positive definite.

Consider the operator R2 → R2given by left-multiplication by the matrix: B =[6 −7,      1 −2]. Find the characteristic polynomial cB(x), the eigenvalues of B, and theeigenspaces of B.

4: Let 6. A = | 0 7 0 0 7 (a) Find a matrix P that diagonalizes A. (b) Find PẬP [using your answer from (a.

in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1