## July 24, 2015

# Characterizing the trace of a matrix

I have been studying for my finals lately, and so I decided to put together a proof of a nice exercise I found in some book. The trace function, given by \(tr : \mathbb{K}^{n \times n} \to \mathbb{K}\), is defined as
tr(A) = \sum_{i=1}^n A_{ii} First of all, the proof of additivity
\begin{equation} \begin{split} tr(A + B) &= \sum_{i=1}^n (A+B)_{ii} \\ &= \sum_{i=1}^n (A)_{ii} + (B)_{ii} \\ &= \sum_{i=1}^n (A)_{ii} + \sum_{i=1}^n (B)_{ii} \\ &= tr(A) + tr(B) \end{split} \end{equation} Afterwards, the proof of homogeneity
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