[Maths Class Notes] on Eigen Values Pdf for Exam

You must have seen in movies where scientists or data analysts come to a certain conclusion by just looking at an ocean of data. It might seem very fictitious but in the real world, it happens in a similar manner. Eigenvalues and eigenvectors of a matrix help in the analysis of financial data and extract information from raw data as well. Eigenvalues are a significant set of scalars that are linked to a linear equation (like matrix equations) and are also known as characteristic roots, real values, and latent roots.

Eigenvalue Definition

An Eigenvalue is a scalar of linear operators for which there exists a non-zero vector. This property is equivalent to an Eigenvector.

What do Eigenvalues Mean?

Geometrically, Eigenvalue meaning is the transformation in the particular point of direction where it is stretched. In case, the Eigenvalue is negative, the direction gets reversed. 

Did You Know?

The time-independent Schrodinger equation in quantum mechanics is an example of an Eigenvalue equation. The wave functions which are associated with the bound states of an electron in a hydrogen atom can be seen as the eigenvectors. The Eigenvalues are interpreted as their energies increasing downward and angular momentum increasing across. A representation of a generalized Eigenvalue problem called Roothaan equations is used in quantum chemistry.

Usage of Eigenvalues

Eigenvalues are used in a wide range of applications and different analyses like stability analysis, vibration analysis, matrix diagonalization, facial recognition, and so on.

Stating an Example

If we consider the equation Ax = λx,  λ is the Eigenvalue there. So, in our Eigenvalue example.

λ = 3 and in case of x = cy, c is a number then. 

What are Eigenvectors?

Eigenvectors are the non zero vectors whose directions aren’t changed even if linear equations are applied. Only a scalar factor can change it. Suppose A is a linear transformation from that of a vector space V and X happens to be the vector in V (non-zero vector) then V is the eigenvector of A in case A(X) is a scalar multiple belonging to X. 

Eigenvalues of eigenvectors are the “axes” which make understanding linear transformations easy. Eigenvalues give the factors by which the stretching or compression and/or flipping of a linear transformation occurs. The more linearly independent eigenvectors associated with a single linear transformation are there the easier it is to understand the linear transformation. 

Eigenvectors and Eigenvalues actually decouple the methods in which the linear transformation acts into a number of independent actions along with separate directions. If the behavior of a linear transformation is obscured due to the choice of basis, it can be made clear if a basis of eigenvectors is chosen. Then the linear transformation is scaling along with the directions of the eigenvectors and the Eigenvalues are the scale factors.

Properties of Eigenvalues

Now let’s look at few of the Properties of Eigenvalues are

• The singular matrix contains zero Eigenvalues.

• Distinct Eigenvalues have a linear dependence on eigenvectors.

• For an Eigenvalue equation, if A is a square matrix, then λ = 0 doesn’t appear to be an Eigenvalue of A. 

Using the properties of Eigenvalues, let’s explain a few Eigenvalues and eigenvectors examples.

Let us find Eigenvalue of matrix from a 2*2 matrix

If  A = [begin{vmatrix}0 &1 \ -2 &-3 end{vmatrix}]

Then the equation becomes,

[[begin{vmatrix}0 &1 \ -2 &-3 end{vmatrix}]] – [[begin{vmatrix}lambda &0 \ 0 &lambda end{vmatrix}]] = 0

[[begin{vmatrix}-lambda &1 \ -2 &-3 – lambda end{vmatrix}]] = [lambda^{2} + 3lambda + 2 = 0]

The two Eigenvalue of matrix we get are

λ1=-1, λ2=-2

Let us find out eigenvector v₁, which is linked with Eigenvalue of matrix

 λ1=-1

A.v₁ = λ₁ . v₁

(A – λ₁) . v₁ = 0

[begin{bmatrix}-lambda_{1} &1 \ -2 &-3 – lambda_{1} end{bmatrix}] . v₁ = 0

[begin{bmatrix}1 &1 \ -2 &-2 end{bmatrix}] . v₁ = [begin{bmatrix}1 &1 \ -2 &-2 end{bmatrix}] . [begin{bmatrix}v_{1,1}\ v_{1,2}end{bmatrix}] = 0

From the equation, we get the following Eigenvalue of matrix= 

[v_{1,1} + v_{1,2} = 0], so

[v_{1,1} = -v_{1,2}]

What is Eigenvalue and What it Really Means?

The most faced question is what is Eigenvalue. It is a nonzero characteristic vector of a linear transformation that changes the maximum when a scalar transformation is applied to it. Geometrically speaking, an Eigenvalue is a scaling factor for the eigenvector. An eigenvector resembling a true non-zero Eigenvalue is the factor by which it is stretched. Suppose T is a linear transformation from a vector space V over a field F into itself. If v is a nonzero vector in V, it can be said that v is an eigenvector of T if T(v) is a scalar multiple of v. 

The geometric multiplicity of an Eigenvalue has to be one at least. In other words, each Eigenvalue must have one associated eigenvector at least. The geometric multiplicity of an Eigenvalue should not exceed its algebraic multiplicity under any circumstance. The study of the actions of associative algebra is termed as the field of representation theory. 

The classical method of calculation is to primarily find the Eigenvalues. This should be followed by the calculation of the eigenvectors for each Eigenvalue. This method is not suited for non – exact arithmetics like the floating – point.