Eigenvectors and Eigenvalues: A deeper understanding
The literal meaning of “Eigen” is characteristic which comes from the German language. These vectors are aptly named as these vectors ultimately give the trend something follows. Any matrix that has an eigenvector, all its coordinates will ultimately follow a relation as close to the eigenvector as possible. To expand on this I will be using some examples.
The Fibonacci Sequence
To explain the Fibonacci sequence, it is just the sum of the previous two terms. If we start the sequence with 0 and 1 the sequence would be
0, 1, 1, 2, 3, 5, 8, 13…
and it goes on till infinity. If we graph it in a Cartesian coordinate system, we see that we can obtain a vector and a square matrix to give the next value in the Fibonacci sequence that is
The first row represents the next value in the sequence while the second row represents the current value in the sequence. If we plot this, we see that each value is approximately 1.618 times the previous value. If we start with 0 and 1, the tenth value would be 55 and the eleventh value would be 89. We can also see that 55 multiplied by 1.618 is also approximately 89. If we calculate the eigenvalues for this vector, they come out to be -0.618 and 1.618. The eigenvectors for this are
Not only will the sequence follow the trend that it will be 1.618 times the previous term but it will also be on the line which follows the vector that is the same eigenvector for the eigenvalue of 1.618 as 1.618 is the dominant eigenvalue. By this we can understand the importance of eigen here as ultimately a sequence if written in the form of a matrix will follow the characteristic defined by its eigenvalues and its eigenvectors. Following this we can easily deduce the value to an increasingly accurate degree for the next term as the term number increases.
This can be done with not only just the starting values 0 and 1, but any values. We will see the sequence following the same eigenvalue as the factor by which the next term will increase. The values will also approach the line that represents the eigenvector in the system as they increase.
Population Growth modelling
In the 1940’s P.H Leslie found a way for modelling population growth with which we can find to a relatively accurate degree the population growth. The Leslie matrix functions on the probability of the rate of reproduction and chance of survival of the particular specie we are dealing with. The Leslie matrix is formed by putting in the respective values in the matrix
where n is the number of stages we have divided the population in, F is the number of females in the population and P is the probability of the members belonging to 1 stage of the population to go to the next. We only take the female population because they are the ones responsible with reproduction in a given population. We substitute this matrix in the matrix equation
xₖ₊₁ = L . xₖ
where xᵏ is the matrix
The value k in this matrix represents the year after which we are dealing with the initial population and n is the number of stages the population is divided in. This equation is obtained by the logic that
x₁ = L . x₀
and
x₂ = L . x₁ = L . (L . x₀) = L² . x₀
and so on. Although with this matrix equation we can easily calculate the population growth but for large values of k, a large number of calculations is required and hence we find the eigenvectors and the corresponding eigenvalues of the system. Through this we get the general trend in which the system is growing and ultimately we can easily calculate the value of xₖ for large values of k.
Solving System of Differential equations
By putting the appropriate values in an augmented matrix, we can solve the system of differential equations with the help of Eigen value and their corresponding Eigen vectors. After first finding the eigen values of the augmented matrix, we arrange them in the form of a diagonal matrix(D) whose diagonal entries are the eigen values. We then find the corresponding eigen vectors which we arrange in a matrix(P).
We will substitute the diagonal matrix in the equation u = Du’, where u is the solution of the system. It should also be noted that D vector shows us by how much the system is changed in every iteration, hence it also represents the rate of change of the system. Once we find u, we put it in the equation y = Pu where P is the matrix containing the eigenvectors. The matrix P will give us the general direction of how the values will change in each iteration and when we will multiply these values we will be able to get the solution of the system(y).
Especially by looking at the last example of system of differential equations, we can see how eigenvectors and eigenvalues show us the general trend of how a system changes. The eigenvalues shows us the magnitude of the rate of change of the system and the eigenvectors shows us the direction that change is taking place in.
