ReviewLet A be n × n with independent eigenvectors v1,, vn.ThenA can be diag onalized as A = PDP−1.Example 1. Diagonalize th e following matri x, if possible.A =2 0 0−1 3 1−1 1 3Solution.Example 2. Suppose A = PDP−1. Then, what is An?Solution.Armin [email protected] differential equationsExample 3. The differential equation y′= ay with initial condition y(0) = C issolvedy(t) = Cea t. (This solution is unique.)Why?Example 4. Our goal is to solve (systems of) differe ntial equations like:y1′= 2y1y2′= −y1+3y2+y3y3′= −y1+y2+3y3y1(0) = 1y2(0) = 0y3(0) = 2In matrix form:Key idea: to solvey′= Ay, introduce eA tDefinition 5. Let A be n × n. The matrix exponential iseA=It shares many proper ties of the usual exponential:• eAis inv er tible and (eA)−1=• eAeB= eA+B= eBeAif AB = BA•ddteA t=• The solution to y′= Ay, y(0) = y0is y =Example 6. If A =2 00 5, then :eA=eA t=Clearly, this works to obt ain eDfor any diagonal matr ix D.Armin [email protected] 7. Suppose A = PDP−1. Then, eA=Why?Example 8. (continued) We wish to solve:y′=2 0 0−1 3 1−1 1 3y , y(0) =121Recall that the solution to y′= Ay, y(0) = y0is y =A = PDP−1with P =1 1 01 0 10 1 1, D =224eA tCheck (optional) that y =indeed solves the original problem:y′=Armin [email protected] 9. Solve the differentia l equationy′=0 11 0y , y(0) =10.Solution.Armin
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