Q4
modle answer but number is different
Find the (possibly complex) Eigen values of the matrix Find the corresponding eigenspaces (which are subspaces of C3). By finding a basis B for C3 consisting of eigenvectors of f : x rightarrow Ax, find a diagonal matrix D with [f]B = D Find a (complex) matrix P such that A = PDP-1, where D is the diagonal matrix in (c), and check your answer by checking that AP = PD. Hint. Let P = PB where D is the basis found in (c). det (A lambda I3)= det So the Eigen values, which are the solutions of det(A - lambda I3) = 0, are 2,-3i,3i. The eigenspace corresponding to 2 consists of those vectors (r, y, z) epsilon C3 such that i.e. -2x + 3z = 0,0 = 0, -3x - 2z = 0. From a Gaussian elimination or simple manipulation x = z = 0. So the eigenspace is {(0, y,0):y epsilon C}. The eigenspace corresponding to 3i consists of those vectors (r, y. z) epsilon C3 such that i.e. -3ix + 3z = 0, (2 - 3i)y = 0, -3x - 3iz = 0. i.e. x = -iz.y = 0. x = - iz. So the eigenspace is {(-iz,0. z) : z epsilon C} Replacing i by -i in the calculation above, the eigenspace corresponding to -3i is {(iz,0,2): z epsilon C}. Let B = {(0,1,0), (-i,0,l),(i,0,l)}. Then f(0,1,0) = (2,0,0) = 2(1,0,0),f(-i,0,1) = (3,0,3i) = 3t(-i,0,l),f(i,0,l) (3,0, -3i) = -3i(t',0,1).