# 2Mathematical Foundations

## 2.1 Matrix Algebra

We may define a **matrix** as an ordered set of elements arranged in a rectangular array of rows and columns. Thus a matrix ** A** may be represented as

where *a*_{ij}, *i* = 1, …, *m*; *j* = 1, …, *n*, is the (representative) element in the *i*th row and *j*th column of ** A**. Since there are

*m*rows and

*n*columns in

**, the matrix is said to be “of order**

*A**m*by

*n*” (denoted (

*m*×

*n*)). When

*m*=

*n*, the matrix is square and will simply be referred to as being “an

*n*th order” matrix. To economize on notation, we may represent

**in the alternative fashion**

*A*Oftentimes we shall need to utilize the notion of a matrix within a matrix, i.e. a **submatrix** is the (*k* × *s*) matrix ** B** obtained by deleting all but

*k*rows and

*s*columns of an (

*m*×

*n*) matrix

**.**

*A*Let us now examine some fundamental matrix operations. Specifically, the **sum** of two (*m* × *n*) matrices ** A** = [

*a*

_{ij}],

**= [**

*B**b*

_{ij}] is the (

*m*×

*n*) matrix

**+**

*A***=**

*B***= [**

*C**c*

_{ij}], where

*c*

_{ij}=

*a*

_{ij}+

*b*

_{ij},

*i*= 1, …,

*m*;

*j*= 1, …,

*n*,

*i.e*, we add corresponding elements. Next, to multiply an (

*m*×

*n*) matrix

**by a scalar**

*A**λ*we simply multiply each element of the matrix by the scalar or

(In view of these operations it is evident that ** A** −

**=**

*B***+ (−1)**

*A***= ...**

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