Eigenvalues, Eigenvectors, Diagonalization, Reduction to the Canonical Form - Engineering Assignment Help

Assignment Task :

Task:

Aims of this laboratory

The aim of this computer laboratory is to illustrate the procedure for diagonalization of a quadratic curve. As it was shown in the lectures, any quadratic curve can be diagonalized (transformed to its canonical form) by using a certain orthogonal transformation matrix. This matrix can always be represented by a rotation matrix. In other words, any quadratic curve can be made diagonal by rotating the curve by a certain angle α either clock- or anti-clockwise. In the lectures we have shown that the eigenvectors of the matrix, associated with the curve, give the directions of the new coordinate axes, in which the curve is diagonal (canonical). The purpose of the following example is to determine such angle α so that the curve becomes diagonal, when it is rotated by α. 1 Quadratic curves The quadratic curve is given by the implicit equation −0.382 x 2 − 1.618 y 2 + 3.80428 xy − 1 = 0. (1) This curve can also be represented in the matrix form 1 = x y −0.3820 1.9021 1.9021 −1.6180 x y . (2) Create a new m-file and proceed as follows:

• Create the matrix A = −0.3820 1.9021 1.9021 −1.6180 (3)

• Find the eigenvalues and eigenvectors of A [M,D]=eig(A)

• Check that the matrix A can be diagonalized using the modal matrix M that consists of the normalized and orthogonal eigenvectors. To this end, check that the matrix transpose(M)*A*M coincides with the spectral matrix D. • Convince yourself that the eigenvectors of A are orthogonal.

• Inspect the eigenvalues of A and make your conclusion about the type of the quadratic curve.

• Next, we want to plot the quadratic curve together with the directions, given by the two eigenvectors of A.

Several plotting commands can be displayed in one figure using the following template: figure hold on plot(...) %% plotting command 1 plot(...) %% plotting command 2 plot(...) %% plotting command 3 hold off Plot the quadratic curve using ezplot(’-0.382*x^2-1.618*y^2+3.80428*x*y-1’,[-3,3]) In the same figure, plot two lines that point in the directions of the two eigenvectors of A. This can be done using the parametric equations of a line x y = t vx vy , t ∈ (−∞,∞), where t is a parameter and vx and vy are the coordinates of the direction vector.

For example, to plot the line that points in the direction of the first eigenvector, simply set vx = M(1, 1) and vy = M(2, 1). • Determine the rotation angle α by comparing the transposed modal matrix MT with the general rotation matrix R(α) = cos(α) sin(α) − sin(α) cos(α) . • Now that you have plotted the curve and the eigenvectors in one figure, convince yourself that the quadratic form can be diagonalized (transformed to its canonical form) by rotating the curve by α = 126o clockwise, or by rotating the curve by α = 54o anti-clockwise. 2 Exercise Given a quafratic curve 2.0528x 2 + 8.9472y 2 − 5.7850xy = 1. • Determine the type of the quadratic curve. • The curve can be diagonalized if it is rotated by angle α clockwise. Determine the smallest possible α in degrees. Round the answer to the unit place.

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