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- Question : 1RE - Obtain the eigenvalues and corresponding eigenvectors of the matrices
- Question : 2RE - Find the principal stress values (eigenvalues) and the corresponding principal stress directions (eigenvectors) for the stress matrix Verify that the principal stress directions are mutually orthogonal.
- Question : 3RE - Find the values of b and c for which the matrix has values of b and c calculate all the eigenvalues and corresponding eigenvectors of the matrix A
- Question : 4RE - Use Gerschgorin
- Question : 5RE - Using the power method find the dominant eigenvalue and the corresponding eigenvector of the matrix starting with an initial vector [1 1 1]T and working to three decimal places. (b) Given that another eigenvalue of A is 1.19 correct to two decimal places, find the value of the third eigenvalue using a property of matrices. (c) Having determined all the eigenvalues of A, indicate which of these can be obtained by using the power method on the following matrices: (i) A_1; (ii) A - 3I.
- Question : 6RE - Consider the differential equations dx
- Question : 7RE - Find the eigenvalues and corresponding eigenvectors for the matrix "8 - 8 -2~ A =4 - 3 -2 3 -4 1 Write down the modal matrix M and spectral matrix A of A, and confirm that M ^AM = A
- Question : 8RE - Show that the eigenvalues of the symmetric matrix 11 Show that the matrix are 9, 3 and -3. Obtain the corresponding eigenvectors in normalized form, and write down the normalized modal matrix M Confirm that M TA M = A where A is the spectral matrix of A.
- Question : 9RE - In a radioactive series consisting of four different nuclides starting with the parent substance N1 and ending with the stable product N4 the amounts of each nuclide present at time t are given by the differential equations model = - 6N1 = 6N - 4N2 dt = 4N2 - 2N3 dt ?N = 2N3 dt Express these in the vector-matrix form N = A N where N = [N1 N2 N3 N4]T. Find the eigenvalues and corresponding eigenvectors of A. Using the spectral form of the solution, determine N4(t) given that at time t = 0, Nt = C and N2 = N3 = N4 = 0.
- Question : 10RE - (a) Given A = use the Cayley-Hamilton theorem to find (i) A7 - 3A6 + A4 + 3A3 - 2A2 + 3I (ii) Ak, where k > 0 is an integer. (b) Using the Cayley-Hamilton theorem, find eAt when
- Question : 11RE - Show that the matrix has an eigenvalue X = 1 with algebraic multiplicity 3. By considering the rank of a suitable matrix, show that there is only one corresponding linearly independent eigenvector e1. Obtain the eigenvector e1 and two further generalized eigenvectors. Write down the corresponding modal matrix M and confirm that M lAM = J, where J is the appropriate Jordan matrix. (Hint: In this example care must be taken in applying the procedure to evaluate the generalized eigenvectors to ensure that the triad of vectors takes the form { T2w, Tw, w}, where T = A - XI, with T2w = e1.)
- Question : 12RE - The equations of motion of three equal masses connected by springs of equal stiffness are x = -2x + y y = x - 2y + z V = y - 2z Show that for normal modes of oscillation x = X coswt, y = Y coswt, z = Z coswt to exist then the condition on X = w2 is Find the three values of X that satisfy this condition, and find the ratios X:Y:Z in each case.
- Question : 13RE - Classify the following quadratic forms: (a) 2x2 + y2 + 2z2 - 2xy - 2yz (b) 3x2 + Vy2 + 2z2 - 4xy - 4xz (c) 16x2 + 36y2 + Hz2 + 32xy + 32xz +16yz (d) -21.x2 + 30xy - 12xz - Uy2 + 8yz - 2z2 (e) -x2 - 3y2 - 5z2 + 2xy + 2xz + 2yz
- Question : 14RE - Show that e1 = [1 2 3]T is an eigenvector of the matrix and find its corresponding eigenvalue. Find the other two eigenvalues and their corresponding eigenvectors. Write down in spectral form the general solution of the system of differential equations dx 2
- Question : 15RE - (a) Find the SVD form of the matrix (b) Use the SVD to determine the pseudo inverse Af and confirm it is a right inverse of A. (c) Determine the pseudo inverse Af without using the SVD.
- Question : 16RE - From (1.51) the unitary matrices U and O and sigma matrix
- Question : 17RE - A linear time-invariant system (A, b, c) is modelled by the state-space equations x(t) = A x(t) + bu(t) y(t) = cTx(t) where x(t) is the n-
- Question : 18RE - A third-order system is modelled by the state-space representation where x = [x1 x2 x3]T and u = [u1 u2]T. Find the transformation x = Mz which reduces the model to canonical form and solve for x(t) given x(0) = [10 5 2]T and u(t) = [t 1]T.
- Question : 19RE - The behaviour of an unforced mechanical system is governed by the differential equation (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. (c) Using the result x(t) = M eJtM-1x(0) obtain the solution to the given differential equation.
- Question : 20RE - (Extended problem) Many vibrational systems are modelled by the vector-matrix differential equation x(t) = A x(t) (1) where A is a constant n x n matrix and x(t) = [x1(t) x2(t) . . . xn(t)]T. By substituting x = eXtu, show that X2u = A u (2) and that non-trivial solutions for u exist provided that | A - X2I | = 0 (3) Let X12, X22, . . . , X2n be the solutions of (3) and u1, u2, ... , un the corresponding solutions of (2). Define M to be the matrix having u1, u2, . . . , un as its columns and S to be the diagonal matrix having X1, X2, ... , X2n as its diagonal elements. By applying the transformation x(t) = M q(t), where q(t) = [q1(t) q2(t) ... to (1), show that & = S q (4) and deduce that (4) has solutions of the form q, = Ci sin(wit + a,) (5) where c, and at are arbitrary constants and X- = jw, with j = V(-1). The solutions X2 of (3) define the natural frequencies w of the system. The corresponding solutions q, given in (5) are called the normal modes of the system. The general solution of (1) is then obtained using x(t) = M q(t). A mass-spring vibrating system is governed by the differential equations x1(t) = -3x1(t) + 2x2(t) x2(t) = x1(t) - 2x2(t) with x1(0) = 1 and x2(0) = i1(0) = x2(0) = 2. Determine the natural frequencies and the corresponding normal modes of the system. Hence obtain the general displacement x1(t) and x2(t) at time t & 0. Plot graphs of both the normal modes and the general solutions.

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