Last modified on 25 August 2009, at 03:17

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.

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 some 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.