Mathematical Methods of Physics/Matrices

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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.

Definition[edit]

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 Transformations[edit]

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 Matrices[edit]

Determinant[edit]

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

Trace[edit]

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

Transpose[edit]

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

Matrix Product[edit]

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

Properties[edit]

(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.