The theory of eigenvalues and eigenvectors is a powerful tool to solve the problems in economics, engineering, and physics. These problems revolve around the singularities of **A** – λ**I**, where λ is a parameter and **A** is a linear transformation. The aim of this chapter is to study the numerical methods to find eigenvalues of a given matrix **A**.

**Definition 4.1.** Let **A** be a square matrix of dimension *n* × *n*. The scalar λ is said to be an eigenvalue for **A** if there exists a non-zero vector **X** of dimension *n* such that **AX** = λ**X**.

The non-zero vector **X** is called the eigenvector corresponding to the eigenvalue λ. Thus, if λ is an eigenvalue for a matrix **A**, then

or

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