## 3

## Real and Complex Matrices

#### 3.1 Introduction

A matrix with real entries is called a real matrix. We have studied various types of real matrices and their properties. Matrices with complex entries are also important in the study of quantum mechanics, quadratic forms, *etc*. We now consider three classes of real square matricesâ€”symmetric, skew-symmetric and orthogonal matrices and also their counterparts with complex elements which are important in applications. But first we define certain terms.

#### 3.2 Orthogonal/Orthonormal System of Vectors

#### 3.2.1 Norm of a Vector

The norm of a vector *X* denoted by || *X* || is defined by

If

*E.g.*

#### 3.2.2 Orthonormal ...