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4. a) i) i., Let H-(-³) 5 7 H= Find the characteristic polynomial of H. Is H diagonalisable? Explain your answer. Use the Cayley-Hamilton theorem to write H-1 and H³ as linear combinations of I and H.

4. a) i) i., Let H-(-³) 5 7 H= Find the characteristic polynomial of H. Is H diagonalisable? Explain your answer. Use the Cayley-Hamilton theorem to write H-1 and H³ as linear combinations of I and H.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 42EQ
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4. a) i) i., iii, L -9 Let H = (- = (-3--²³) 5 combinations of I and H. V= Find the characteristic polynomial of H. Is H diagonalisable? Explain your answer. Use the Cayley-Hamilton theorem to write H-1 and H³ as linear b) i) [---] Show that the matrix A = (12) has only one eigenvalue and that is the corresponding eigenvector. ii)] Solve the system of differential equations with initial conditions (0) = 2, y(0) = -3. r' (t)=-4y+ 3et y' (t) = 9x + 12y-4e6t; U PRO Let F = FX-XF. You may assume without proof that the map f is linear. i) [ -- Prove that f²(X) = -2FXF, i.e., prove that f(f(x)) = −2FXF. Prove that ƒ³ (X) = 0, for any X € M2,2(R), i.e., prove that f(f(f(x))) = 0, for any Xe M2,2 (R). Let - (8) Define the map f: M2,2(R) → M2,2(R) by f(X) = be the standard basis in M2,2 (R). A) Find 1 = E11 - (₁). ₂- (₂2). ₂ - (2), E₂-(2). E12 = E21 = f(E1), f(E12), f(E21), f(E22). B) Find the matrix, M, of the map f with respect to the standard basis in the domain and in the co-domain. C) Without computing, explain why M² #0 and M³ = 0. D) Let J be the Jordan Normal form of the matrix M, i.e., MJ. Explain why 0 is the only eigenvalue of the matrix J (and of M). E) Find the Jordan normal form of the matrix M.
Transcribed Image Text:4. a) i) i., iii, L -9 Let H = (- = (-3--²³) 5 combinations of I and H. V= Find the characteristic polynomial of H. Is H diagonalisable? Explain your answer. Use the Cayley-Hamilton theorem to write H-1 and H³ as linear b) i) [---] Show that the matrix A = (12) has only one eigenvalue and that is the corresponding eigenvector. ii)] Solve the system of differential equations with initial conditions (0) = 2, y(0) = -3. r' (t)=-4y+ 3et y' (t) = 9x + 12y-4e6t; U PRO Let F = FX-XF. You may assume without proof that the map f is linear. i) [ -- Prove that f²(X) = -2FXF, i.e., prove that f(f(x)) = −2FXF. Prove that ƒ³ (X) = 0, for any X € M2,2(R), i.e., prove that f(f(f(x))) = 0, for any Xe M2,2 (R). Let - (8) Define the map f: M2,2(R) → M2,2(R) by f(X) = be the standard basis in M2,2 (R). A) Find 1 = E11 - (₁). ₂- (₂2). ₂ - (2), E₂-(2). E12 = E21 = f(E1), f(E12), f(E21), f(E22). B) Find the matrix, M, of the map f with respect to the standard basis in the domain and in the co-domain. C) Without computing, explain why M² #0 and M³ = 0. D) Let J be the Jordan Normal form of the matrix M, i.e., MJ. Explain why 0 is the only eigenvalue of the matrix J (and of M). E) Find the Jordan normal form of the matrix M.
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