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.