proof of properties of trace of a matrix

Let us check linearity. For sums we have

$\displaystyle\operatorname{trace}(A+B)$	$\displaystyle=$	$\displaystyle\sum\limits_{i=1}^{n}(a_{i,i}+b_{i,i})\,\,\,\,\,\,\,\,\,\,\,\mbox% {(property of matrix addition)}$
	$\displaystyle=$	$\displaystyle\sum\limits_{i=1}^{n}a_{i,i}+\sum\limits_{i=1}^{n}b_{i,i}\,\,\,\,% \mbox{(property of sums)}$
	$\displaystyle=$	$\displaystyle\operatorname{trace}(A)+\operatorname{trace}(B).$

Similarly,

$\displaystyle\operatorname{trace}(cA)$	$\displaystyle=$	$\displaystyle\sum\limits_{i=1}^{n}c\cdot a_{i,i}\,\,\,\,\,\mbox{(property of % matrix scalar multiplication)}$
	$\displaystyle=$	$\displaystyle c\cdot\sum\limits_{i=1}^{n}a_{i,i}\,\,\,\,\,\mbox{(property of % sums)}$
	$\displaystyle=$	$\displaystyle c\cdot\operatorname{trace}(A).$

2.

The second property follows since the transpose does not alter the entries on the main diagonal.

The proof of the third property follows by exchanging the summation order. Suppose $A$ is a $n\times m$ matrix and $B$ is a $m\times n$ matrix. Then

$\displaystyle\operatorname{trace}AB$	$\displaystyle=$	$\displaystyle\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}A_{i,j}B_{j,i}$
	$\displaystyle=$	$\displaystyle\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{n}B_{j,i}A_{i,j}\,\,\,\,% \mbox{(changing summation order)}$
	$\displaystyle=$	$\displaystyle\operatorname{trace}BA.$

The last property is a consequence of Property 3 and the fact that matrix multiplication is associative;

$\displaystyle\operatorname{trace}(B^{-1}AB)$	$\displaystyle=$	$\displaystyle\operatorname{trace}\big{(}(B^{-1}A)B\big{)}$
	$\displaystyle=$	$\displaystyle\operatorname{trace}\big{(}B(B^{-1}A)\big{)}$
	$\displaystyle=$	$\displaystyle\operatorname{trace}\big{(}(BB^{-1})A\big{)}$
	$\displaystyle=$	$\displaystyle\operatorname{trace}(A).$

Title	proof of properties of trace of a matrix
Canonical name	ProofOfPropertiesOfTraceOfAMatrix
Date of creation	2013-03-22 13:42:54
Last modified on	2013-03-22 13:42:54
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	4
Author	Daume (40)
Entry type	Proof
Classification	msc 15A99