MAT 315: TEST 3. Problem 1. Note that in this problem the matrix is not diagonalizable and has only two different eigenvalues. It is a multipart problem in which we consider the following 3 × 3-matrix 2 A= 0 −4 1 2 −12 0 0 . 3 Part A. Find the two eignvalues λ1 and λ2 and fill out the following table Eigenvalues λ1 = λ2 = AlgebraicM ultiplicity alg.mult(λ1 ) = alg.mult(λ2 ) = Eigenvectors vλ1 = vλ2 = GeometricM ultiplicity geo.mult(λ1 ) geo.mult(λ2 ) Part B. Use the table above to determine the Jordan normal form J of the matrix A. Part C. One of the eigenvalues above has a different algebraic multiplicity then geometric multiplicity. This means we need to find a generalized eigenvector q. To this aim, use the matrix equation AQ = QJ to find this vector q. Part D. Calculate eJt and uses this to calculate eAt = QeJt Q−1 by using your answer in Part C. Part E. (Extra credit) Consider the non-homogouns system of differential equad tions dt v = Av + b(t) where b(t) = (1, t, 0). Calculate the particular soloution R At vp = e u(t) by first computing u(t) = e−tA b(t)dt + C where C = (c1 , c2 , c3 ). Part F. (Extra credit) Use the initial condition v(0) = (1, 2, −1)T to solve the initial value problem in Part E and write down the resulting particular solution vp . Problem 2. Find the Jordan"normal decomposition A = QJQ−1 to calculate # √ 13 −19 . A of the following matrix A = 9 −48 1 2 MAT 315: TEST 3. Problem 3. Consider the following matrix 2 1 0 A = 0 1 5 . 0 −2 3 Part A. Calculate the two Jordan blocks J1 and J2 of A. One of them is a complex Jordan block, say J2 " cos(θ) Part B. Write J2 in polar coordinate form J2 = r2 · − sin(θ) sin(θ) cos(θ) # Part C. Find the eigenvectors of A and put them in a transition matrix Q with real number entries. Part D. How does A = QJQ−1 act on the vector space R3 . Problem 4 Briefly explain why one of the following facts are true. A) Every symmetric m atrix is diagonalizable. B) Every linear system of homogenous linear ordinary differential equations has a unique solution. C) The determinant of a matrix is the product of all of its eigenvalues counting multipilicity. D) Every n × n matrix has at least one eigenvector. 2 E) The standard nomral curve f (x) = √12π e−x is an eigenfunction of the Fourier R∞ transform F(f (x))(s) = −∞ f (x)e−isx dx. Problem 5 Part A. Calculate N 0 , N 1 , N 2 , N 3 of the 0 1 N = 0 0 0 0 following matrix 0 1 0 Part B. Calculate the matrix polynomial p(t) = I + N t + N 2 t2 /2 + N 3 t3 /6 Part C. Calculate (2I + N )0 , (2I + N )1 , (2I + N )2 /2, (2I + N )3 /6 Part D. Approximate eJt where J = 2I + N by calculating by using Part C. P3 k=0 (2I + N )k /k! Part E. Verify directly the claim in Part D by calculating eJt by using the formula for exponentials of matrices.