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.

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

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.

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.

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

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.

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.

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

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

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

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

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.

FAQ (Frequently Asked Questions)

1. What are the Eigenvalue Applications?

Answer: In geology, especially in glaciology, eigenvectors and Eigenvalues are used to summarize a mass of information of the orientation and dip of a clast fabric’s constituents. Eigenvectors use singular value decomposition for the compression of images. Eigenvalues and eigenvectors are also applied in the cases of stability and vibrational analysis as well as facial recognition and matrix diagonalization.

We use eigenvectors daily without even realizing it. For instance, while having food our mind automatically formulates the ingredients' ratios and transforms the tastes into its principal components of sour, bitter, sweet, and such other tastes. In other words, Eigenvalues and eigenvectors help us to understand or perceive things very effectively.

2. What is the Importance of Eigenvalues or Eigenvectors?

Answer: Since now we know what is Eigenvalue, let's discuss a few benefits of Eigenvalues or eigenvectors.

Eigenvectors are the most efficient set of basis functions for describing the variability of data. They are often used in reducing the dimension of large data sets after the selection of a few modes with significant Eigenvalues and to find new uncorrelated variables. Thus they are very helpful for the least-square regressions of the systems which are badly conditioned.

It can be said that Eigenvalues work in characterizing important properties of linear transformations, for example, whether a system of linear equations has a unique solution. Eigenvalues also describe the physical properties of a mathematical model in many cases.