Eigenvalues And Eigenvectors

The ‘Eigen’ is adopted from the German term Eigen which means own, characteristics, or proper. Earlier, it was utilised for studying principal axes of rigid bodies’ rotational motion. Today, 

Eigenvalues and Eigenvectors both have a number of applications. Some of them include vibration analysis, facial recognition, stability analysis, matrix diagonalisation, atomic orbitals, and many more. Here, we’ll learn more through this elaborated introduction to Eigenvalues and Eigenvectors. 

One of the simple illustrations of these entities is the original painting of the Mona Lisa. Here, every point present in the painting is represented in the form of a vector pointing from the painting’s centre to that point. The linear transformation we can observe in the painting is called shear mapping. 

The article shares a detailed history and definition of the Eigenvalues And Eigenvectors and looks at the important questions that you must know regarding the topic.

Introduction To Eigenvalues And Eigenvectors: History 

As per Eigenvalues and Eigenvectors notes, Eigenvalues are observed in the context of matrix theory or linear algebra. Originally, they were used for the study of differential equations and quadratic forms. During the 18th century, the rigid body’s rotational motion was studied by Leonhard Euler. He established the significance of the principal axes. Later, Joseph-Louis Lagrange discovered that the inertia matrix’s Eigenvectors are the principal axes. 

During the beginning of the 19th century, another term was coined by Augustin-Louis Cauchy, the characteristic root (Racine caractéristique). Now, it is known as the Eigenvalue. 

After many other discoveries related to Eigenvalues and Eigenvectors, in 1929, the 1st numerical algorithm came to the surface to compute them in a power method published by Richard von Mises.

Define Eigenvalue 

After a detailed introduction to Eigenvalues and Eigenvectors, let us see Eigenvalues and Eigenvectors’ meanings. Eigenvalues are associated with the linear equation system as the special scalars set. It has its general use in the area of matrix equations. We can also call it characteristic root, latent roots, or proper values. Eigenvalue acts as a scalar that transforms the Eigenvector.

The equation used is Ax = λx.

λ = number or scalar value.

λ is the Eigenvalue of A.

Eigenvalue exists for each real matrix. Here, the fundamental algebra theorem = Eigenvalue for a complex matrix. 

Define EigenVectors

These are the non-zero vectors that don’t change their directions on the application of any linear transformation. Instead, it can change only by a scalar factor. 

Here, Suppose A acts as a linear transformation from V (vector space) and x (non zero) acts as a vector in V. Then, V will be an Eigenvector of A if A(X) acts as a scalar multiple of x.

In, Ax = λx

x acts as an Eigenvector of A and corresponds to λ, which is the Eigenvalue of A.

Note:

• Infinite Eigenvectors can correspond to 1 Eigenvalue.  
• The Eigenvectors linearly depend on different Eigenvalues.

Important Eigenvalues Properties

• Eigenvalues of hermitian and real symmetric matrices tend to be real.
• Eigenvalues of real skew hermitian and skew-symmetric matrices can either be zero or pure imaginary.
• Eigenvalues of orthogonal and unitary matrices show unit modulus |λ| = 1.
• If A has Eigenvalues as λ1, λ2 …….. λn, then kA will have Eigenvalues as kλ1, kλ2……. kλn.
• If A has Eigenvalues as λ1, λ2 ……. λn, then A-1 will have Eigenvalues as 1/λ1, 1/λ2 ……. 1/λn.
• If A has Eigenvalues as λ1, λ2 ……… λn, then Ak will have Eigenvalues as λ1k, λ2k ……. λnk.
• Eigenvalues of A are equal to the Eigenvalues of AT, the transpose matrix.
• Eigenvalues sum is equal to the A’s diagonal elements sum or Trace of A 
• Eigenvalues’ product is represented as |A|
• A maximum number of different Eigenvalues of A is the size of A.
• If A and B are the same order matrices, then AB’s Eigenvalues will be equal to BA’s Eigenvalues.

Conclusion

The introduction to Eigenvalues and Eigenvectors teaches us that complex linear operations can be solved in a simplified manner. These have been a great help in segregating the complex linear problems in the simplest terms, helping to get results much more efficiently. The article here talked about the German origin of the terms and their history of how they got coined. Next, we went through a detailed definition of Eigenvalues and Eigenvectors to further learn about them. Finally, ensure to go through the mentioned properties of Eigenvalues to understand how it is used in matrix theory.