Matlab Tutorial 5: Linear Equations

It is time to try the solution. Write the following in Matlab:

b=     A=
  8    3 -2  4
 -5    5  8 -6
-17    9 -2  7

a= if we plot this, it will give a straight line. See figure 1b.

>> X=inv(A)*b

Figure 1b

It is not necessary always to adapt the data to a linear function, but in this example the data seemed to fit nicely for a first order function. For other data we might have looked for a quadratic, cubic or a fourth-order polynomial.

y=a₁x² + a₂x + a₃ , quadratic function
y=a₁x³ + a₂x² + a₃x + a₄ , cubical function

Eigenvalues and eigenvectors

Eigenvalue problems can be found within many different fields. One purpose could be to locate cracks in building constructions. we know that a building has its own natural frequencies (eigenvalues) if there are no cracks, but the frequencies will change if cracks appear. The same principle can be used in oil drilling. By listening to the ground you will get different answers (frequencies) depending on whether the ground contains oil, mud, water or something else. Different linear systems mean different eigenvalues.

We can find eigenvalue problems in cars, ships, airplanes, bridges and so on. These issues are of course not limited to mechanical systems, but can also be found in any kind of system modeled by linear equations, including political, economic, biological, electrical or whatever systems. They could in fact have an identical model description and the same eigenvalues.

Yet another example could be the rotation of the earth. In such a system there will be a fixed direction, namely the axis of the rotation. There must be a north and a south pole and thus a rotation axis. This becomes the eigenvector. An eigenvector specifies the direction. How can we define these?

Let A be a nxn matrix. The X vector will be separated from the zero vector, thus AX=λX , for some number λ . X is called the eigenvector to the matrix A with eigenvalue λ. Eigenvalues and eigenvectors can be complex. A nxn matrix has always n eigenvalues (poles). Eigenvalues can be determined by: det(A-λI)=0.

This equation is commonly known as the characteristic equation (numerator in the transfer function = 0), and when this is solved you get the eigenvalues. For each and every eigenvalue one can decide the eigenvectors by (A-λI)X=0 , where I is the identity matrix. Note that the eigenvalues are closely related to the system, but this is not true for the eigenvectors. They change depending on how we describe the system.

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