Before we get into the properties of Eigenvalues, let us have an overview of Eigenvalues. 

Eigenvalues are frequently introduced in the subjects of linear algebra and matrix theory. Previously, however, they were used in the study of quadratic forms and differential equations. 

Eigenvalues and Eigenvectors are regularly introduced to students in linear algebra courses focused on matrices. So, to understand Eigenvalues, you first need to understand what matrices are. 

What are matrices?

A matrix, if simplified, can be called a transformation. It transforms the coordinate system from one to another. 

A matrix has columns and rows that contain elements. Matrices are the main part of linear algebra and help study linear equations and transformations. 

Matrices are part of a larger concept called tensors, which are extensively used in theoretical physics, like the Einstein Field Equations. 

Matrices are utilised in problems related to machine learning to represent a major set of information. Eigenvectors and Eigenvalues are mainly about setting up one vector with one value to represent a bigger matrix. 

Let’s understand Eigenvalues

To understand the meaning of Eigenvalue properties, let us learn about Eigenvalues first. Eigenvalues are known to be the special scalars set linked with the linear equations system. We use it mainly for matrix equations. 

The word Eigen is German in origin and means characteristic or proper. Thus, we also call Eigenvalues characteristic roots, characteristic values, and proper values. To describe it simply, it is a scalar utilised to transform the Eigenvector. 

Here’s the Eigenvalue equation: 

Ax = λx

Where λ is the scalar value, which is an Eigenvalue of A

In mathematics, an Eigenvector correlates with the real non-zero Eigenvalues pointing in the direction that is stretched due to the transformation. An Eigenvalue is addressed as a factor through which the stretch took place. The transformation’s direction is also negative in the scenario where the Eigenvalue has a negative value. For each real matrix, an Eigenvalue is present. 

Furthermore, the presence of the Eigenvalue is equal to the fundamental algebra theorem in the case of complex matrices.

Significant Eigenvalue properties 

Here are some essential Eigenvalue properties: 

• Eigenvalues are real for hermitian and real-symmetric matrices. 
• Eigenvalues tend to be either 0 or entirely imaginary for real-skew symmetric/skew hermitian matrices. 
• Eigenvalues are of unit modulus |λ| = 1 for orthogonal/unitary matrices. 
• In the case where λ1, λ2…λn are defined as the Eigenvalues for ‘A’, the Eigenvalues for kA would be defined as kλ1, kλ2…kλn.
• In the case where, λ1, λ2…λn are referred to as the Eigenvalues for A, 1/λ1, 1/λ2…1/λn would represent the Eigenvalues for A-1.
• In the case where, λ1, λ2…λn represent Eigenvalues for A, λ1k, λ2k…λnk would be the Eigenvalues for Ak. 
• Eigenvalues of transpose AT are equivalent to A’s Eigenvalues. 
• The diagonal elements’ sum or the trace of matrix A is equivalent to the Eigenvalues’ sum. 
• |A| represents the Eigenvalues’ product.
• The size of matrix A is equivalent to A’s maximum numbers of different Eigenvalues. 
• In the scenario where A and B represent two distinct matrices having the same order, the Eigenvalues of matrices AB are equivalent to the Eigenvalues of matrices BA. 

Having a good understanding of Eigenvalue properties will help you in solving relevant problems more successfully. 

Conclusion

Think of Eigenvectors and Eigenvalues as providing a concise summary of a large matrix. 

Eigenvalues and Eigenvectors are used to simplify the complexity of data. They both can assist us in improved efficiency in computationally complex tasks. 

Eigenvalues and Eigenvectors form the basics of mathematics and computing. The major potential of theoretical and practical applications of Eigenvectors and Eigenvalues is constantly increased further in computing science, allowing every respective calculation of large matrices. This leads to new prospects in theoretical and applied research.