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Matrix Algebra - Tutorial 6

Matrix Algebra - Tutorial 6. 32. Find the eigenvalues and eigenvectors of the following matrices (note the eigenvalues are integers):. Suppose the eigenvectors of A are denoted  1 and  2 and let X = [  1  2 ]. Find X -1 A X. Comment.

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Matrix Algebra - Tutorial 6

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  1. Matrix Algebra - Tutorial 6 32. Find the eigenvalues and eigenvectors of the following matrices (note the eigenvalues are integers): Suppose the eigenvectors of A are denoted 1 and 2 and let X = [12]. Find X-1 A X. Comment. When evaluating the eigenvectors of B, show that the three equations represented by (B-lI)x = 0 are linearly dependent. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  2. 33. The Stochastic Matrix equation below shows how the values of R, C and I change over time. R,C and I are percentages. Use eigenvalue techniques to show that their values will stabilise if this equation is repeatedly applied and hence find their steady value. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  3. 34. In the rotational system above, the angular position of the mass is q, its angular velocity is w = d/dt, and it can be shown that -kq -Fw = Jdw/dt Express these equations as two state equations in q and w if J = 2kgm2, F = 6Nm per rad/s and k = 4Nm per rad. Find the general response of q and w and the particular response if at time 0, w = 0 rad/s and q = 1 rad. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  4. 35. The above circuit can be modelled by equations Use eigenvalue techniques to find the general response of v2 and i1 when L = 4H, R = 8W and C = 0.25F, and the particular response if at time t = 0, i1 = 0 and v2 = 2V. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  5. 36. Consider the following, being a permanent magnet armature controlled d.c. motor in a feedback loop with controller C. The command input is 0. Let q be the motor position and w its speed. Question continued.. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  6. Examining the block diagram, the equations describing the system are w = dq/dt and Tdw/dt = -Cq - w. Express the above in terms of the state variables w and q and find the complex eigenvalues and eigenvectors to the system if T = 0.5s and C = 5. Hence write down the general solution to the system. Find the particular solution if at time t = 0,  = 0 rad/s and  = 3 rad. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  7. Answers EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  8. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

  9. EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

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