Question

3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that the matrix B = 0 1 is not diagonalizable. 0 4] Lo 5. Let 2, and 1, be two distinct eigenvalues of a matrix A (2, # 12). Assume V1, V2 are eigenvectors of A corresponding to 11 and 22 respectively. Prove that V1, V2 are linearly independent.

Answer 1

eigenspaces for the Corresponding and (Ou 3 find all the eigenvalues matrix B=10 21 det 18-00) = 10-d = 0 -d(i-d)+2=0 -d + 2 + 2 = 0 d d +2=0 d = !+ J1 - 4xlx2 = d= 14 F di = LAST d2= 1-i (eigenvalues) find eigenspace for die tri Consider (A-diT) K =0 " ad=1tri 4-571 K1 +222 = 0 -> -)- JFL x, = -2K2 -Kit 2 - LIiKg =0 HUFIK = 4K2 di = 14 F : RTE [ K2 ] ( lt hij eigenspace. Elgen space for d2 = 1-5i consider (A-d2Ijk zo (- / + 7 i 2 TKI - 1 + 7 i K, + 2K2 = 0 + (1-7 k, = 44 kg -K; + $+ fți Ke =0 kg - da = 1-1 : [ 3 ] = { 1-ri eigen space. 7 4 7 buth show that the mathix B=11 l ol is not diagonalizable. 0 1 004) Given matrix B is upper tercinguar matuis. so its eigenvalues are its foctuet of diagonal entries. de = 1 da=1 d₃=4 Two eigenvalues are same. so Given matrix B is not diagonalizable for diagonalizablet all eigenvalues should be distinct).