Complex numbers from the eigenvector and the eigenvalue, Mathematics

Assignment Help:

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


Related Discussions:- Complex numbers from the eigenvector and the eigenvalue

Linear algebra, i have question like proof, can you please help me on it?

i have question like proof, can you please help me on it?

Volume of solids, find the volume of a rectangular based right pyramid with...

find the volume of a rectangular based right pyramid with its base 18 cm by 24 cm and the slanted edge 39 cm

Calculus (The squeeze theorem), When finding the limit as x approaches 0 th...

When finding the limit as x approaches 0 the for function (square root of x^3 + x^2) cos(pi/2x) would the limit not exist because there would be a zero in the denominator?

Profits and loss, what does 1000/q in the ATC equation represent economical...

what does 1000/q in the ATC equation represent economically?

Price cutter sold 85 beach towels what were the total sales, Price Cutter s...

Price Cutter sold 85 beach towels for $6.95 each. What were the total sales? You must multiply the number of towels sold through the price of each towel; 85 × $6.95 = $590.75.

Equations of planes - three dimensional spaces, Equations of Planes Ear...

Equations of Planes Earlier we saw a couple of equations of planes.  Though, none of those equations had three variables in them and were actually extensions of graphs which we

Think smarter, compare: 643,251; 633,512; and 633,893. the answer is 633,51...

compare: 643,251; 633,512; and 633,893. the answer is 633,512. what is the question?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd