(1) Let us consider the 2 x 2 matrix A = (1 2₂¹) G (a) Find eigenvalues of A, show that it is not diagonalizable without

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1 Let Us Consider The 2 X 2 Matrix A 1 2 G A Find Eigenvalues Of A Show That It Is Not Diagonalizable Without 1 (52.9 KiB) Viewed 42 times

(1) Let us consider the 2 x 2 matrix A = (1 2₂¹) G (a) Find eigenvalues of A, show that it is not diagonalizable without computing eigenvectors explicitly. (b) Suppose that v2 = (1,0) and set v₁ = (A-AI) v2 where A is the eigenvalue find in previous problem. Show that A =P ( ₁₂ √) P²² with first column of P is v₁ and second column of P is U2, so we can conjugate A to its Jordan block of size 2. (c) From here show that A2 - 2A + I = O where I is an identity matrix and O is the zero matrix so A satisfies its own characteristic equation.