**Theorem 2.1.1. Equivalent Statements**

If *A* is an *n* × *n* matrix and *λ* is a complex number, then the following are equivalent


Some coefficients of the characteristic polynomial of *A* have a specific shape. The following theorem gives the information about it.

#### **Theorem 2.1.2.**

**2. The linear eigenvalue problem**

has nontrivial solution **x**, where *A* ∈ *C*(*n*,*n*)

limited to the right eigenvectors, which are earlier defined.

principle of Poincare, minmax principle of Courant-Fischer)

spectrum of *A*, and is denoted as *σ*(*A*).

This section is organized as follows:

invertibility, diagonalization)

**5.** General linear eigenvalue problem

equations, as we will see in the following discussion.

or by inserting an identity matrix *I* equivalently

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

**4.** Physical Background

**2.1. Basic properties**

system

58 Applied Linear Algebra in Action

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

an **eigenvector** of *A* **corresponding** to *λ.* The set of all eigenvalues of matrix *A* is called the

The literature discusses the right and left eigenvectors. In our deliberations, we have been

**1.** Basis properties (characteristic polynomial, bases for eigenspaces, eigenvalues and

**2.** QR Algorithm (The QR algorithm is used for determining all the eigenvalues of a matrix. Today, it is the best method for solving the unsymmetrical eigenvalue problems.)

**3.** Mathematical background for Hermitian (symmetric) case (Rayleigh quotient, min max

*A B* **x x** = l

*A I* **x x** = l

( ) *A I* - = l

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

(1)

**.** The scalar *λ* is called **an eigenvalue** of *A,* and **x** is

(2)

(3)

**x 0**. (4)

*A***x= x**l

> If *A* is an *n* × *n* matrix, then the characteristic polynomial *p*(*λ*) of *A* has degree *n*, the coefficient of *λ<sup>n</sup>* is (−1)*<sup>n</sup>*, the coefficient of *λ<sup>n</sup>* − 1 is (−1)*<sup>n</sup>* − 1 *trace*(*A*) and the constant term is *det*(*A*), where *trace*(*A*) := *a*11 + *a*22 + ⋯ + *ann*.

In some structured matrices, eigenvalues can be read as shown in Theorem 2.1.3.

### **Theorem 2.1.3.**

If *A* is an *n* × *n* triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of *A* are entries of the main diagonal of *A.*

Cayley-Hamilton's theorem is one of the most important statements in linear algebra. The theorem states:

## **Theorem 2.1.4.**

Substituting the matrix *A* for λ in characteristic polynomial of *A*, we get the result of zero matrix i.e., *p*(*A*) = 0.

There are a number of methods for determining eigenvalue. Some methods allow finding all the eigenvalues and the other just a few of the eigenvalues. Methods based on first determining the coefficients of the characteristic polynomial, and later to determining the eigenvalue solving algebraic equations are rarely implemented, because they are numerically unstable. In fact, for the coefficients of the characteristic polynomial burdened with rounding errors, and due to numerical instability cause large errors in the eigenvalue. Because of that, the charac‐ teristic polynomial has mainly theoretical significance. The methods, which are based on the direct application characteristic polynomial, are applied in practice only when the character‐ istic polynomial is well conditioned. Also for some structured matrices, we can apply the method for the characteristic polynomial, but we don't calculate directly characteristic polynomial coefficients. The following example describes a class of such matrices.

**Example 2.1.1** The Example of structured matrix which achieve the characteristic polynomial for determining the eigenvalue are Toeplitz matrix. Toeplitz matrix marked as *Tn*, are matrices with constant diagonals. If the Toeplitz matrix is symmetric and positive definite, recursive relation is *pn*(*λ*) = *pn* − 1(*λ*)*β<sup>n</sup>* − 1(*λ*), where are *pn* i *pn* − 1 characteristic polynomial matrix *Tn* i *Tn* − **<sup>1</sup>** respectively a *β<sup>n</sup>* − 1 Schur-Szegö parameter for Yule-Walker system. The above recursive relation enables work with characteristic polynomial without individual accounts of his odds. More information can be found at [1]

The following definitions are introducing two important terms: the geometric multiplicity of *λ*0 and the algebraic multiplicity of *A.*

The eigenvectors corresponding to λ are the nonzero vectors in the solutions space of (*A* − *λI*)**x** = **0.** We call this solution space the **eigenspace** of *A* corresponding to λ.
