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

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 sum 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 $$\delta_{ij}$$, 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.