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EIGENVALUES AND EIGENVECTORS

Eigenvalue is the set of values for a linear differential equation or linear algebraic equation for which it provides a non-zero solution. It is a scalar function.

The eigenvalue is a scalar function that provides a non-zero solution in the case of linear algebraic equations. It can be determined from the characteristics equation of a given matrix. The equation from which the eigenvalue of a matrix can be determined by equating the determinant value of A–λI=0. The values of λ are considered to be the eigenvalues of the function. On the other hand, Eigenvectors are the vector set of non-zero vectors that are associated with a linear equation. The equation from which the Eigenvectors set can be calculated is AI=λI, where A is a scalar matrix, the I is an identity matrix, and λ is the scalar multiple.

Eigenvalues: Overview

The eigenvalue is a mathematical scalar value that is defined to calculate linear algebraic equations. The Eigenvalue of the matrix is a powerful tool that is used to measure the linear dependency of a matrix. The eigenvalues are calculated from the characteristic equation which is defined as det(A-SI)=0, Where A is the scalar matrix, and S is the eigenvalues. For the different eigenvalues, the characteristics equation should be followed. eigenvalues are used to determine the stability of a point in a particular domain.

Eigenvectors and their importance

Eigenvectors play a significant role in reducing the complexity of the linear algebraic equation. It can be calculated from the characteristic’s equation. For each set of vectors, the characteristic equation needs to be zero. It can be defined as a vector set that determines the linear dependence of the equation set.

Relationship of Eigenvalue and Eigenvectors

  • It has been observed that if the Eigenvalues of an equation set are linearly dependent then the Eigenvectors would also be linearly dependent on each other.
  • If the Eigenvalues are linearly independent then the eigenvector would also be linearly independent.
  • For a specific eigenvalue, there would be a specific eigenvector.

Importance of Cayley-Hamilton theorem

Cayley-Hamilton theorem states that every matrix which is squared by nature (Number of rows = number of col.) satisfies its own characteristic equation. It is a proven fact that if an equation satisfies its characteristic equation, then the eigenvalue would be equal to the scalar value of the matrix and the characteristic equation can be replaced by the matrix equation.

Calculation of eigenvalues and Eigenvectors

  • The first step of calculating the eigenvalues is to check whether the matrix is square or not. The Eigenvalues can only be determined for square matrices.
  • The second step of finding the eigenvalues is the construction of [A-λI]
  • The third step is the calculation of the determinant value of [A-λI]
  • In the final step the value of the determinant equals zero and calculates the values of λ. The values of λ are the eigenvalues of that equation set.
  • The Eigenvectors can be calculated with the help of eigenvalues. The equation of finding the eigenvector set is [A-λI]=[0].

Properties of Eigenvalue

  • It has been observed that for scalar, diagonal, and triangular matrices the eigenvalues are nothing but their diagonal elements.
  • The sum of the eigenvalues is equal to the trace (sum of principal diagonal element) of the matrix.
  • The product of the eigenvalues is equal to the determinant value of the matrix.
  • The number of zero eigenvalues is equal to the number of linearly dependent rows or columns of the matrix.
  •  The number of non-zero eigenvalues is equal to the number of rows or columns which are linearly independent.
  • The number of non-zero eigenvalues determine the rank of the matrix and the number of zero eigenvalues determines the nullity of the matrix.

Conclusion

Eigenvalues and eigenvectors are the important mathematical tools of linear algebra that are used to solve linear algebraic equations. The calculation of eigenvalues is performed from the characteristic equation. The concept of eigenvalues and eigenvectors only exists for square matrices. Eigenvalue and eigenvectors are widely used in control systems, communication systems, and designing the stereo system.