By Maxime Bocher

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4. (a) What are the four upper Jordan canonical forms for a 4 x 4 matrix A having complex eigenvalues? (b) What is the form of the solution of the initial value problem (4) in each of these cases? 5. (a) List the seven upper Jordan canonical forms for a 5 x 5 matrix A with a real eigenvalue A of multiplicity 5 and give the corresponding deficiency indices in each case. 8. Jordan Forms 49 (b) What is the form of the solution of the initial value problem (4) in each of these cases? 6. Find the Jordan canonical forms for the following matrices (a) A = 1 0 0 1 2 0 1 2 3 1 0 0 (b) A = -1 2 0 1 0 (c) A = 10 (d) A = 10 (e)A=I1 (f) A= I 1 (g) A= 0 (h) A= I 0 2 1 2 2 1 0 2 1 2 2 1 0 2 0 0 0 2 0 0 2 3 0 2 3 4 0 0 0 2 0 0 0 2 0 1 0 2 1 4 0 2 0 0 1 0 2 0 0 2j 1 4 0 2 1 -1 0 2 0 2 0 1 Find the solution of the initial value problem (4) for each of these matrices.

In this section we establish the fundamental fact that for xo E R" the initial value problem x = Ax x(0) = xo (1) has a unique solution for all t E R which is given by x(t) = eAtXO. (2) Notice the similarity in the form of the solution (2) and the solution x(t) = eatxo of the elementary first-order differential equation i = ax and initial condition x(0) = xo. 4. The Fundamental Theorem for Linear Systems 17 In order to prove this theorem, we first compute the derivative of the exponential function eAt using the basic fact from analysis that two convergent limit processes can be interchanged if one of them converges uniformly.

Using Gaussian reduction, we next find that b2 = 3 and {V(1),v12)3 V21)} with V21) = v21) spans Ker(A - AI)2. Similarly we find b3 = 4 and we must find 63 - 62 = 1 solution of (A - al)v = V(2) where V12) = v12). The third row of this equation implies that x2 = 0 and the remaining equations are then equivalent to x1 +x3+x4 = 0. 8. Jordan Forms 47 Then B = {v( 3), v23), v33), v43>} _ {Vi1), V12), Vi31, V21)} is a basis for Ker(A - \I)3 = R. The matrix P = [v1 v4], with v1 = V(1), V2 = Vi21, v3 = vi31 and v4 = V21), and its inverse are then given by 1 P -1 0 -1 1 -1 0 0 1 1 0 0 0 1 00 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 1 P-1- and 0 respectively.

### An Introduction to the Study of Integral Equations by Maxime Bocher

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