Mathematical Methods of Physics/Matrices

We have already, in the previous chapter, introduced the concept of matrices as representations for linear transformations. Here, we will deal with them more thoroughly.

DefinitionEdit

Let   be a field and let  , . An n×m matrix is a function  .

We denote  . Thus, the matrix   can be written as the array of numbers  

Consider the set of all n×m matrices defined on a field  . Let us define scalar product   to be the matrix   whose elements are given by  . Also let addition of two matrices   be the matrix   whose elements are given by  

With these definitions, we can see that the set of all n×m matrices on   form a vector space over  

Linear TransformationsEdit

Let   be vector spaces over the field  . Consider the set of all linear transformations  .

Define addition of transformations as   and scalar product as  . Thus, the set of all linear transformations from   to   is a vector space. This space is denoted as  .

Observe that   is an   dimensional vector space

Operations on MatricesEdit

DeterminantEdit

The determinant of a matrix is defined iteratively (a determinant can be defined only if the matrix is square).

If   is a matrix, its determinant is denoted as  

We define,  

For  , we define  

We thus define the determinant for any square matrix

TraceEdit

Let   be an n×n (square) matrix with elements  

The trace of   is defined as the sum of its diagonal elements, that is,

 

This is conventionally denoted as  , where  , called the Kronecker delta is a symbol which you will encounter constantly in this book. It is defined as

 

The Kronecker delta itself denotes the members of an n×n matrix called the n×n unit matrix, denoted as  

TransposeEdit

Let   be an m×n matrix, with elements  . The n×m matrix   with elements   is called the transpose of   when  

Matrix ProductEdit

Let   be an m×n matrix and let   be an n×p matrix.

We define the product of   to be the m×p matrix   whose elements are given by

  and we write  

PropertiesEdit

(i) Product of matrices is not commutative. Indeed, for two matrices  , the product   need not be well-defined even though   can be defined as above.
(ii) For any matrix n×n   we have  , where   is the n×n unit matrix.