Decide which of the matrices A in Exercises 1 through 20 are diagonalizable. If possible, find an invertible S and a diagonal D such that S~lAS = D. Do not use technology.

MATH 1220 Notes for Week #7 22 February 2016 WarmUp ● Remind yourself of the of the statement of the Intermediate Value Theorem (IVT): ○ on a closed interval [a, b] where a < b and f(a) < 0, f(b) > 0, and f is continuous, there exists a number z such that f(z) = 0 ○ note: z is not unique; example is f(x) = sin(x) because sin(x) = 0 on the period kπ Equilibrium Theory Suppose f : [0, 1] → [0,1]. f(0) / 0 and f(1) / 1 . Then there exists a real number c in [0, 1] such that f(c) = c. This is called a fixed point of f . If you overlay the graph y = x , the graph of f cannot satisfy the previously laid out properties without crossing y = x where all points are fixed points. Proof: ● Define a new function h(x) =