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 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

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

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

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.