Skip to main content

Mathematical Methods of Physics: For Undergraduate

Sunil Karna

Subsection 2.3.6 The Trace of a Matrix

The algebric sum of the elements of principal diagonal in any square matrix is called the trace of a matrix, i.e.,
\begin{equation*} Tr A = \sum\limits_{k} a_{kk}, \end{equation*}
For example:
\begin{equation*} A=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \vdots & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \end{equation*}
then trace of \(A = Tr A = a_{11}+a_{22}+ \cdots + a_{nn} = \sum\limits_{k=1}^{n} a_{kk}\text{.}\)

Properties.

  1. Cyclic Theorem: The trace of a product of two matrices is independent of the order of multiplication, i.e., \(Tr (AB) = Tr(BA)\text{.}\)

    Proof.

    We have -
    \begin{equation*} Tr (AB) = \sum\limits_{i} (AB)_{ij} = \sum\limits_{i}\left\{\sum\limits_{j} a_{ij}b_{ji}\right\} \end{equation*}
    \begin{equation*} = \sum\limits_{j}\sum\limits_{i}a_{ij}b_{ji} = \sum\limits_{j}\sum\limits_{i}b_{ji}a_{ij} = \sum\limits_{j}(BA)_{ji} = Tr(BA) \end{equation*}
    This holds even if \(AB \neq BA\text{.}\) It can be generilezed to \(Tr(ABCD)= Tr(DCBA)\text{.}\)
  2. Trace of the product of a symmetric and an anti - symmetric matrix is zero.

    Proof.

    Let us consider that A is a symmetric and B is an anti - symmetric matrix,
    \begin{equation*} A=A^{t} \quad \text{and}\quad B=-B^{t} \end{equation*}
    then,
    \begin{equation*} Tr (AB) = \sum\limits_{i}(AB)_{ij} = \sum\limits_{i}\sum\limits_{j}a_{ij}b_{ji} = \sum\limits_{i}\sum\limits_{j}(-b_{ij})a_{ji} \end{equation*}
    [since, \(A = A'\) and \(B = B'\)]
    \begin{equation*} = \sum\limits_{i}\sum\limits_{j}(b_{ij})a_{ji} = - \sum\limits_{i}(BA)_{ii} = -Tr (BA) = -Tr (AB) \end{equation*}
    or, \(2Tr(AB) = 0\) or, \(Tr(AB) =0.\)
PrevTopNext