Before we begin, we'll introduce some terminology. When we examine the flight dynamics of anaircraft, we're usually concerned with three types of rotation: pitch, roll, and yaw.Pitchis the angle ofrotation associated with the rise and fall of the nose of the aircraft, either pointing up or down. Rollisthe angle by which the wings deviate from being level, so that one wing rises up and the other dropsdown. Yawrefers to rotation around a vertical axis, moving the nose of the airplane left or right; achange in yaw results in a change of heading for the plane. Take a look at the animated imagesbelow, and try to identify the three different kinds of rotation.These images are provided by NASA and are in the public domain.We will consider a model used in the design of commercial aircraft. This is a differential equation thatdescribes the effect of rate of change of rudder angle on the rate of change in yaw. The yaw ratemodelis the 4×4 system

12/3/20154 | Systems of ODEs13/16(9)x′(t) = Ax(t) + Bu(t)with a single function uas the driving term. Here, uis the rate of change of rudder angle, and the fourcomponents of xare:x1=lateral (side-to-side) velocity;x2=yaw rate;x3=roll rate;x4=pitch rate.Ais the 4×4 matrix , and Bis the 4×1 matrix In our exercise we will ignore the effects of the driving term and instead consider the homogeneoussystem x′= Ax. In effect, this means our aircraft has no control system. We'll see very soon that suchan airplane couldn't fly for long.

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Exercise 4.5a. What are the eigenvalues and eigenvectors of the matrix A? Include your input commandsand your output in your Word document.b. According to the mathematical definitions, is the system x′= Axstable, asymptoticallystable, or unstable? Why?c. One of the eigenvalues you obtained is very close to zero. Look at the correspondingeigenvector. Which component is biggest? Based on that, which type of rotation is this