2 - Vectors, Matrices, and Tensors

2.1 Introduction
2.2 Introduction to Matrices
2.3 Vector Spaces
2.4 Definition of Tensors
2.5 The Symmetric Eigenproblem
2.6 The Rayleigh Quotient and the Minimax Characterization
2.7 vector and Matrix Norms
2.8 Exercises


2.1 Introduction

The use of vectors, matrices, and tensors is of fundamental importance in engineering analysis because it is only with the use of these quantities that the complete solution process can be expressed in a compact and elegant manner.

From a simplistic point of view, matrices can simply be taken as ordered arrays of numbers that are subjected to specific rules of addition, multiplication, and so on. It is of course important to be thoroughly familiar with these rules, and we review them in this chapter.

2.2 Introduction to Matrices

The effectiveness of using matrices in practical calculations is readily realized by considering the solution of a set of linear simultaneous equations.

Definition : A matrix is an array of ordered numbers. A general matrix consists of mn numbers arranged in m rows and  n columns.

We say that this matrix has order m x n (m by n). When we have only one row (m=1) or one column (n=1), we also call A a vector.