Method of Eigenvalues and Eigenvectors
Solved Problems
Example 1.
Find the general solution of the system of differential equations
Solution.
We calculate the eigenvalues
In this example, the auxiliary equation has two distinct real roots.
Find an eigenvector
Let the eigenvector
After matrix multiplication we get a system of two equations:
Both equations are linearly dependent. From the second equation we find the relation between the coordinates of the eigenvector:
Similarly, we find the second eigenvector
We get the system of two identical equations:
From this we find the coordinates of the eigenvector
Hence,
Thus, the system has two different eigenvalues and two eigenvectors. The general solution is given by
where
Example 2.
Find the general solution of the system of differential equations
Solution.
We seek a solution of the form
where
We obtain two eigenvalues as a pair of complex conjugate numbers. Find the eigenvector
Both equations are linearly dependent. From the second equation we have:
Thus, the eigenvector
Consequently, the complex number
Transform the exponential function by Euler's formula:
The solution
or after multiplication
In the complex solution, the real and imaginary parts are linearly independent. Separating them, we find the general solution:
Thus, the general solution has the form
where
Example 3.
Find the general solution of the system of differential equations
Solution.
The matrix of the system is diagonal:
Therefore, we can just say that the eigenvectors are
However, we will construct a solution, following the general algorithm. Calculate the eigenvalues of the matrix
The matrix has a single eigenvalue with algebraic multiplicity
It is clear that any nonzero vector
Note that we have the case where the eigenvalue
The general solution of the system of equations can be written as
Example 4.
Find the general solution of the system of differential equations
Solution.
We calculate the eigenvalues of
Expand the determinant along the first column:
You may notice that one of the roots of the cubic equation is the number
The quadratic equation, in turn, has roots
Now we define an eigenvector for each of the eigenvalues.
Determine the vector
Denoting
As a result, we have a system of linear algebraic equations:
In this system, the first and third equations are the same, i.e., the rank of the system is
Thus, the eigenvector
where for simplicity we set
Similarly, we find the coordinates of the second eigenvector
Let
Hence, the eigenvector
At
Now we calculate the coordinates of the third eigenvector
We choose
Substituting
Thus, the eigenvector ({\mathbf{V}_3}) has coordinates
The general solution can be written as
where
Example 5.
Find the general solution of the system
Solution.
Let's start with finding the eigenvalues of the matrix
Factoring the left side, we get
It can be seen that the auxiliary equation has one real and two complex roots (as a pair of complex conjugate numbers):
The eigenvector
After multiplying we get:
We see that the rank of the system of equations is
So the first eigenvector has coordinates
Consider now the pair of complex conjugate roots
and identify the real and imaginary parts, which will represent two linearly independent solutions. By implementing this plan, we write the matrix-vector equation for the vector
We get the following system of equations:
Transform the first equation into a more convenient form, multiplying it by
Get rid of the complex numbers in the denominators of the coefficients:
Then the first equation becomes:
Let's go back to the system of equations and reduce it to a triangular form to determine its rank:
Transform the second equation:
Here the coefficient of the variable
Hence, the second equation is
Similarly, we transform the third equation:
Calculate the coefficient of
Then the third equation can be written as
that is, it coincides with the second equation.
So, the rank of the system is
We take as an independent variable
Thus, we determined the eigenvector
Now we construct the solution
Represent
Hence,
The calculated real and imaginary parts of the complex vector solution
where
Example 6.
Find the general solution of the system of differential equations
Solution.
Determine the eigenvalues of the given matrix:
It can be noted that the cubic equation has a root
The roots of the quadratic equation are:
The initial matrix of the system is symmetric. So it will have three eigenvectors. This means that the algebraic and geometric multiplicity of the root
Find the eigenvectors corresponding to the number
We see that all three equations are identical. Leaving one equation, and choosing
It follows that the coordinates of the first eigenvector (with
Accordingly, the coordinates of the second linearly independent eigenvector (when
Now we define the third eigenvector
Here we choose as a free variable
Hence, the eigenvector
The general solution of the system of differential equations is given by
where