Complex numbers from the eigenvector and the eigenvalue, Mathematics

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Complex numbers from the eigenvector and the eigenvalue.

Example1: Solve the following IVP.

2144_Complex numbers from the eigenvector and the eigenvalue.png

We first require the eigenvalues and eigenvectors for the given matrix.

1679_Complex numbers from the eigenvector and the eigenvalue1.png

= l2 + 27

l1,2 = + 3 √(3i)

Therefore, now that we have the eigenvalues recall which we only need to determine the eigenvector for one of the eigenvalues as we can determine the second eigenvector for free from the first eigenvector as:

l1 =  3 √(3i),

We have to to solve the subsequent system.

199_Complex numbers from the eigenvector and the eigenvalue2.png

By using the first equation we find,

(3 - 3 √(3i)) h1-  9h2 = 0,

h2 = 1/3 (1 - (√(3i))) h1

Therefore, the first eigenvector is,

144_Complex numbers from the eigenvector and the eigenvalue3.png

h1 = 3

While finding the eigenvectors during these cases ensures that the complex number appears in the numerator of any fractions as we'll require this in the numerator later on.  Also attempt to clear out any fractions by suitably picking the constant. It will make our life simple down the road.

 Here, the second eigenvector is,

585_Complex numbers from the eigenvector and the eigenvalue4.png

Though, as we will see we won't require this eigenvector.

The solution which we get from the first eigenvalue and eigenvector is,

452_Complex numbers from the eigenvector and the eigenvalue5.png

Therefore, as we can notice there are complex numbers in both the exponential and vector that we will require to get rid of in order to use that as a solution. Recall from the complex roots section of the second order differential equation section which we can use Euler's formula to find the complex number out of the exponential. Doing it, we get

2396_Complex numbers from the eigenvector and the eigenvalue6.png

The subsequent step is to multiply the cosines and sines in the vector.

61_Complex numbers from the eigenvector and the eigenvalue7.png

Here combine the terms along with an "i" in them and split such terms off from those terms that don't include an "i". Also factor the "i" out of that vector.

1030_Complex numbers from the eigenvector and the eigenvalue8.png

= u?(t) +v?(t)

Here, it can be demonstrated as u?(t) and v?(t)are two linearly independent solutions to the system of differential equations. It means that we can utilize them to form a general solution and both they are real solutions.

Therefore, the general solution to a system along with complex roots is,

x? (t) = c1u?(t) +c2v?(t)

Here u?(t) and v?(t)are found by writing the first solution as:

x? (t) = u?(t) + i v?(t)

For our system so, the general solution is,

1330_Complex numbers from the eigenvector and the eigenvalue9.png

We now require applying the initial condition to it to find the constants,

32_Complex numbers from the eigenvector and the eigenvalue10.png

This leads to the subsequent system of equations to be solved,

3c1 = 2;

c1 + √3c2 = -4;

By solving both equations we get:

c1 = (2/3) and c2 = (14/3√3)

The actual solution is, so,

557_Complex numbers from the eigenvector and the eigenvalue11.png


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