DETERMINANTS AND EIGENVALUES 1. Introduction Gauss ...

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106 II. DETERMINANTS AND EIGENVALUESThenExample 1. LetHence,A = t[ ]0 1.−1 0[ ]A 2 = t 2 −1 00 −1[ ]A 3 = t 3 0 −11 0[ ]A 4 = t 4 1 00 1[ ]A 5 = t 5 0 1−1 0.[ ] [ ]e A 1 0 0 1= +t + 1 [ ]−1 00 1 −1 0 2 t2 + 1 [ ]0 −10 −1 3! t3 +...1 0[ 1−t 2=2 + t4t34!− ... t−3! + t55! − ... ]−t+ t33! − t55! + ... 1− t2 2 + t4[ ]4! − ...cos t sin t=.− sin t cos tAs in the example, a series of n × n matrices yields a separate series for eachof the n 2 possible entries. We shall say that such a series of matrices converges ifthe series it yields for each entry converges. With this rule, it is possible to showthat the series defining e A converges for any n × n matrix A, but the proof is abit involved. Fortunately, it is often the case that we can avoid worrying aboutconvergence by appropriate trickery. In what follows we shall generally ignore suchmatters and act as if the series were finite sums.The exponential function for matrices obeys the usual rules you expect an exponentialfunction to have, but sometimes you have to be careful.(1) If 0 denotes the n × n zero matrix, then e 0 = I.(2) The law of exponents holds if the matrices commute, i.e., if B and C aren × n matrices such that BC = CB, then e B+C = e B e C .(3) If A is an n × n constant matrix, then d dt eAt = Ae At = e At A. (It is worthwriting this in both orders because products of matrices don’t automaticallycommute.)Here are the proofs of these facts.(1) e 0 = I +0+ 1 2 02 +···=I.
6. THE EXPONENTIAL OF A MATRIX 107(2) See the Exercises.(3) Here we act as if the sum were finite (although the argument would work ingeneral if we knew enough about convergence of series of matrices.)ddt eAt = d (I + tA + 1 dt2 t2 A 2 + 1 3! t3 A 3 + ···+ 1 )j! tj A j +...=0+A+ 1 2 (2t)A2 + 1 3! (3t2 )A 3 + ···+ 1 j! (jtj−1 )A j + ...=A+tA 2 + 1 2 t2 A 3 1+ ···+(j−1)! tj−1 A j + ...=A(I +tA + 1 2 t2 A 2 1+ ···+(j−1)! tj−1 A j−1 + ...)=Ae At .Note that in the next to last step A could just as well have been factored out onthe right, so it doesn’t matter which side you put it on.Exercises for Section 6.[ ]λ 01. (a) Let A = . Show that0 µe At =[ ]eλt00 e µt .⎡⎤λ 1 0 ... 00 λ(b) Let A = ⎢ 2 ... 0⎣.⎥. ....⎦ . Such a matrix is called a diagonal matrix.0 0 ... λ nWhat can you say about e At ?[ ]0 02. (a) Let N = . Calculate e1 0Nt .⎡(b) Let N = ⎣ 0 0 0⎤1 0 0⎦. Calculate e Nt .0 1 0⎡⎤0 0 ... 0 01 0 ... 0 0(c) Let N be an n × n matrix of the form0 1 ... 0 0⎢⎥⎣. . ... . .⎦ . What is the0 0 ... 1 0smallest integer k satisfying N k = 0? What can you say about e Nt ?[ ]λ 03. (a) Let A = . Calculate e1 λAt . Hint: use A = λI +(A−λI). Note thatA and N = A − λI commute.
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