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1. [8 marks] Write equation 7x{ +6x1x2 + 7x3 = 1 as a matrix-vector quadratic form,...
Quadratic forms arise often in geometry, physics and engineering and it is desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:Q = 3x12 + 5 x22 + 3x32 - 2x2 x3 + 2x1 x3 - 2 x1 x2 to a standard form by making an approximate change in variables X = MY , where M is an orthogonal matrix. i) Write Q as X T AX . ii) Find the Eigenvalues of A. Is the quadratic form...
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e. the matrices of unit eigenvectors) of the corresponding transformations and write down explicite expressions for canonical cOordinates (y1, 2, y3) in terms of the original coordinates (x1, X2, X3). State what surfaces these quadratic forms correspond to = > (a) x + 4x1r2 + 4a13-8a2x3 = 1; (b) a3a3a^ + 4xj2 +4x131223 1; (c) 4a7 2a2 2axjx2 2x13+ 6x23 = 1....
Question 10 Find the matrix of the quadratic form associated with the quadratic equation 7x2 +1&xy – 2y2 -32=0. Selected Answer a
for a matrix solution of the quadratic (3) Find a formula of the form x = -B C equation ax2 + bx +c = 0. Here c denotes and 0 denotes 0 0 (Hint: First show how the square root of any number D can be obtained using a where it looks different depending matrix of the form on whether D is negative. Then use the quadratic formula.) positive or for a matrix solution of the quadratic (3) Find a...
Find the matrix A of the quadratic form associated with the equation. 6x2 - 9xy - 6y2 + 5 = 0 A= Find the eigenvalues of A. (Enter your answers as a comma-separated list.) 2 = Find an orthogonal matrix P such that PTAP is diagonal. (Enter the matrix in the form [[row 1], [row 2], ...), where each row is a comma-separated list.) P = Need Help? Read It Talk to a Tutor
a) Write the augmented matrix for the linear system that corresponds to the matrix equation AX = b b) solve the above system c) write the solution as a vector A= 0 4 5 -1 3 2 and b= 1 24 5 4 2
Consider an undamped system where the vector-matrix form of the system model is: [F(t) [8 orë Mx + x = LO 183, 2000 -1800 x (-1800 45001 The system is initially at rest with X (0) = 0 and 2,0)=0 when input F(t) = 84 sin 15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two...
Consider an undamped system where the vector-matrix form of the system model is: F(t) [8 olx Mx + Kx = 0 18X, + 2000 -1800 x -1800 4500 I:1-[0] The system is initially at rest with x(0) = 0 and x,0)=0 when input F(t) = 84 sin15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two...
1. [10 marks] Consider the quadratic equation x2-0.03x 3.99. a) Write this as a perturbation problem. Is it sin (b) Letting-ao + aie+ a2ε2 + . . . solve up to and including terms of order 0(e2). (c) Compare the solution with the "exact" solution. 1. [10 marks] Consider the quadratic equation x2-0.03x 3.99. a) Write this as a perturbation problem. Is it sin (b) Letting-ao + aie+ a2ε2 + . . . solve up to and including terms of...
Please show all steps for how to solve this perturbation problem. Thank you. 1. [10 marks] Consider the quadratic equation 2 -0.03x 399 (a) Write this as a perturbation problem. Is it singular or regular? (b) Letting x = ao + aie+ a2e2 + . . . solve up to and including terms of order 0(e2). (c) Compare the solution with the "exact" solution. 1. [10 marks] Consider the quadratic equation 2 -0.03x 399 (a) Write this as a perturbation...