Computing for Numerical Methods Using Visual C++

Chapter 9. Eigenvalues and Eigenvectors

• 9.1 Eigenvalues and Their Significance

• 9.2 Exact Solution and Its Existence

• 9.3 Power Method

• 9.4 Shifted Power Method

• 9.5 QR Method

• 9.6 Visual Solution: Code9

• 9.7 Summary

• Numerical Exercises

• Programming Challenges

EIGENVALUES AND THEIR SIGNIFICANCE

The eigenvalues of a square matrix are important parameters in determining things like the stability of the structure where the matrix is based. A bridge depends on the strength of its beam across a given length whose stability is determined by its eigenvalues. In image processing, the eigenvalues of a matrix that represent an image hold the key to the quality of the image. With a proper technique, a blur image can be transformed into a crisp one once the eigenvalues of its corresponding matrix are known.

Therefore, finding the eigenvalues of a square matrix and their corresponding eigenvectors has become an important problem in many applications. The eigenvalue problem can be stated as follows:

Given a matrix A, find the nonzero vectors v such that Av = λ v, where λ is the eigenvalue and v is its corresponding eigenvector. The real eigenvalues and its eigenvectors of the matrix will not exist if no value of v can satisfy Av = λv.

In many applications, it may not be necessary to find all the eigenvalues of a given matrix. In this case, the eigenvalues whose values are extreme become the focus. An eigenvalue whose modulus is the largest is called the most dominant eigenvalue, whereas one that is smallest is called ...