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 $M=\{1,2,\dots ,m\}$,$N=\{1,2,\dots ,n\}$. 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 $A=\left(\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1m}\\ a_{21}& a_{22}& a_{23}& \dots & a_{2m}\\ a_{31}& a_{32}& a_{33}& \dots & a_{3m}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& a_{n3}& \dots & a_{nm}\end{array}\right)$

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 $b_{ij}=c{a}_{ij}$. 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 $(T_1+T_2)𝐮=T_1𝐮+T_2𝐮$ and scalar product as $(cT)𝐮=c(T𝐮)$. 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, $\left|\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\right|=a_{11}{a}_{22}-a_{21}{a}_{12}$

For $n=3$, we define $\left|\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)\right|=a_{11}\left|\left(\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right)\right|-a_{12}\left|\left(\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right)\right|+a_{13}\left|\left(\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right)\right|$

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 $tr(A)=\sum _{i,j=1}^n{a}_{ij}{\delta }_{ij}$, where ${\delta }_{ij}$, called the Kronecker delta is a symbol which you will encounter constantly in this book. It is defined as

${\delta }_{ij}=\{\begin{array}{cc}1,& \text{if }i=j\\ 0,& \text{if }i\ne j\end{array}$

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 $a_{ij}^T=a_{ji}$

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.