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

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

Consider the set of all n×m matrices defined on a field ${\displaystyle F}$. Let us define scalar product ${\displaystyle cA}$ to be the matrix ${\displaystyle B}$ whose elements are given by ${\displaystyle b_{ij}=ca_{ij}}$. Also let addition of two matrices ${\displaystyle A+B}$ be the matrix ${\displaystyle C}$ whose elements are given by ${\displaystyle c_{ij}=a_{ij}+b_{ij}}$

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

## Linear Transformations

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

Define addition of transformations as ${\displaystyle (T_{1}+T_{2})\mathbf {u} =T_{1}\mathbf {u} +T_{2}\mathbf {u} }$ and scalar product as ${\displaystyle (cT)\mathbf {u} =c(T\mathbf {u} )}$. Thus, the set of all linear transformations from ${\displaystyle U}$ to ${\displaystyle V}$ is a vector space. This space is denoted as ${\displaystyle L(U,V)}$.

Observe that ${\displaystyle L(U,V)}$ is an ${\displaystyle mn}$ 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 ${\displaystyle A}$ is a matrix, its determinant is denoted as ${\displaystyle |A|}$

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

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

We thus define the determinant for any square matrix

### Trace

Let ${\displaystyle A}$ be an n×n (square) matrix with elements ${\displaystyle a_{ij}}$

The trace of ${\displaystyle A}$ is defined as the sum of its diagonal elements, that is,

${\displaystyle tr(A)=\sum _{i=1}^{n}a_{ii}}$

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

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

The Kronecker delta itself denotes the members of an n×n matrix called the n×n unit matrix, denoted as ${\displaystyle I}$

### Transpose

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

## Matrix Product

Let ${\displaystyle A}$ be an m×n matrix and let ${\displaystyle B}$ be an n×p matrix.

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

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

### Properties

(i) Product of matrices is not commutative. Indeed, for two matrices ${\displaystyle A,B}$, the product ${\displaystyle BA}$ need not be well-defined even though ${\displaystyle AB}$ can be defined as above.
(ii) For any matrix n×n ${\displaystyle A}$ we have ${\displaystyle AI=IA=A}$, where ${\displaystyle I}$ is the n×n unit matrix.