6.3 4. The matrix is 0 -1 1 2 I will use x instead of lambda. The characteristic polynomial is x^2 -2x + 1 = (x-1)^2. 1 is the only eigenvalue. Ker (I - A) = the kernel of 1 1 -1 -1 = solutions of x+y = 0, i.e. { -y : y in R} y which has basis -1 1 Since the geometric multiplicity (1) is smaller than the algebraic multiplicity (2), there is no eigenbasis. 6. The matrix is 2 3 4 5 with char. polynomial x^2 - 7x - 2. By the quadratic formula, the eigenvalues are (7 +/- sqrt{57})/2 where +/- means plus or minuis and sqrt{57} means the square of 57. There will be a one-dimensional eigenspace for each eigenvalue in this case. The eigenspace for (-7 + sqrt{57})/2 is the kernel of (7 + sqrt{57})/2 - 2 -3 -4 (7 + sqrt{57})/2 - 5 or (3 + sqrt{57})/2 -3 -4 (-3 + sqrt{57})/2 (The second row IS a mutliple of the first.) Then u = 3 (3 + sqrt{57})/2 is an eigenvector. Entirely similarly, v = 3 (3 - sqrt{57})/2 is an eigenvector for the eigenvalue (-7 - sqrt{57})/2 and u, v give an eigenbasis. 18. The matrix is 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 The characteristic polynomial is x^2(x - 1)^2, and the eigenvalues are 0, 1, both with algebraic multiplicity two. The eigenspace for 0 is the kernel of the original matrix, with has basis e_1, e_3. The eigenspace for 1 is the kernel of 1 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 Since the corresponding equations give x_1 = -x_4 = x_3 = 0, the kernel is one-dimensional and has basis e_2, the only eigenvector for the eigenvalue 1. Since the geometric multiplicity is less than the algebraic multiplicity, there is no eigenbasis. 6.4 22. The matrix is 1 3 -4 10 The characteristic polynomial is x^2 + 11x + 22. The eigenvalues happen to be real: (-11 +/- sqrt{121 - 88})/2 = (-11 =/- sqrt{33})/2 24. The matrix is 0 1 0 0 0 1 5 -7 3 The characteristic polynomial is x -1 0 det 0 x -1 = -5 7 x-3 x(x(x-3) - (-1)(7)) - (-1)(0(x-3) - (-1)(-5)) = x(x^2 - 3x + 7) +(0 - 5) = x^3 - 3x^2 + 7x - 5. By inspection, x = 1 is one root, and we can factor out x-1: we get (x-1)(x^2 - 2x + 5). The roots of the quadratic are (2 +/- sqrt(4-20))/2 or 1 +/- 2i which, together with 1, give the eigenvalues. 28. Call the two complex conjugate eigenvalues a+bi, a-bi. Since the real eigenvalue is 2 and the trace and determinant are 8 and 50, resp. we know that the sum of the eigenvalues is 8 and the product is 50. This gives 2 + 2a = 8 or a = 3, and 2(a^2 + b^2) = 50 and so 9 + b^2 = 25, from which b = 4 or -4. This implies that the complex eigenvalues are 3 +/- 4i.