Mathematical Methods of Physics/Matrices

< Mathematical Methods of Physics

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.

DefinitionEdit

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 TransformationsEdit

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 MatricesEdit

DeterminantEdit

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

TraceEdit

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

TransposeEdit

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 ProductEdit

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

PropertiesEdit

(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.