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Mathematical Methods of Physics/Matrices

< Mathematical Methods of Physics

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.



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


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


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  


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  


(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.