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.

Definition

Let F be a field and let $M=\{1,2,\ldots,m\}$,$N=\{1,2,\ldots,n\}$. An n×m matrix is a function $A:N\times M\to F$.

We denote A(i,j) = a_ij. Thus, the matrix A can be written as the array of numbers $A=\begin{pmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1m} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2m} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \ldots & a_{nm} \\ \end{pmatrix}$

Consider the set of all n×m matrices defined on a field F. Let us define scalar product cA to be the matrix B whose elements are given by b_ij = ca_ij. Also let addition of two matrices A + B be the matrix C whose elements are given by c_ij = a_ij + b_ij

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

Linear Transformations

Let U,V be vector spaces over the field F. Consider the set of all linear transformations $T:U\to V$.

Define addition of transformations as $(T_1+T_2)\mathbf{u}=T_1\mathbf{u}+T_2\mathbf{u}$ and scalar product as $(cT)\mathbf{u}=c(T\mathbf{u})$. Thus, the set of all linear transformations from U to V is a vector space. This space is denoted as L(U,V).

Observe that L(U,V) is an mn dimensional vector space

Operations on Matrices

Determinant

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

If A is a matrix, its determinant is denoted as | A |

We define, $\left| \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{pmatrix}\right| =a_{11}a_{22}-a_{21}a_{12}$

For n = 3, we define $\left| \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{pmatrix}\right|=a_{11}\left| \begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{pmatrix}\right|-a_{12}\left| \begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \\ \end{pmatrix}\right|+a_{13}\left| \begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \\ \end{pmatrix}\right|$

We thus define the determinant for any square matrix

Trace

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

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

$tr(A)=\sum_{i=1}^n a_{ii}$

This is conventionally denoted as $tr(A)=\sum_{i,j=1}^na_{ij}\delta_{ij}$, where δ_ij, called the Kronecker delta is a symbol which you will encounter constantly in this book. It is defined as

$\delta_{ij} = \left\{\begin{matrix} 1, & \mbox{if } i=j \\ 0, & \mbox{if } i \ne j \end{matrix}\right.$

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

Transpose

Let A be an m×n matrix, with elements a_ij. The n×m matrix A^T with elements $a_{ij}^T$ is called the transpose of A when $a^T_{ij}=a_{ji}$

Matrix Product

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

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

$c_{ij}=\sum_{k=1}^n a_{ik}b_{kj}$ and we write C = AB

Properties

(i) Product of matrices is not commutative. Indeed, for two matrices A,B, the product BA need not be well-defined even though AB can be defined as above.

(ii) For any matrix n×n A we have AI = IA = A, where I is the n×n unit matrix.