**2. The linear eigenvalue problem**

This section considers the linear eigenvalue problem of finding parameter λ such that the linear system

$$\mathcal{A}\mathbf{x} = \mathcal{A}\mathbf{x} \tag{1}$$

has nontrivial solution **x**, where *A* ∈ *C*(*n*,*n*) **.** The scalar *λ* is called **an eigenvalue** of *A,* and **x** is an **eigenvector** of *A* **corresponding** to *λ.* The set of all eigenvalues of matrix *A* is called the spectrum of *A*, and is denoted as *σ*(*A*).

The literature discusses the right and left eigenvectors. In our deliberations, we have been limited to the right eigenvectors, which are earlier defined.

This section is organized as follows:


$$
\mathbf{A}\mathbf{x} = \mathbf{\hat{\lambda}}\mathbf{B}\mathbf{x} \tag{2}
$$

#### **2.1. Basic properties**

In this section we outline the basic concepts and theorems, which will allow us to understand further elaboration. The eigenvalue problem is related to the homogeneous system of linear equations, as we will see in the following discussion.

To find the eigenvalues of *n* × *n* matrix *A* we rewrite (1) as

$$A\mathbf{x} = \lambda I\mathbf{x} \tag{3}$$

or by inserting an identity matrix *I* equivalently

$$\mathbf{x}\left(\mathcal{A} - \lambda I\right)\mathbf{x} = \mathbf{0}.\tag{4}$$

Such system is homogeneous system and the system (3) has a nontrivial solution if and only if

$$\det\left(\mathcal{A} - \lambda I\right) = 0.\tag{5}$$

This is called the **characteristic equation** of *A;* the scalars satisfying this equation are the eigenvalues of *A*. When is expanded, the determinant det(*A* − *λI*) is polynomial *p* in λ, and it is called the **characteristic polynomial** of *A*.

The following theorem gives the link between the characteristic polynomial of the matrix *A* and its eigenvalues.
