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.

Contents

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.