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| Title of object: |
trace of a matrix |
| Canonical Name: |
TraceOfAMatrix |
| Type: |
Definition |
| Created on: |
2001-11-16 22:30:34-05 |
| Modified on: |
2002-03-12 23:19:40-05 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
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Content:
\textbf{Definition:} \newline Let $A$ be a square matrix of
dimension $n$. The trace of the matrix is the sum of the main
diagonal.\begin{center} $\operatorname{trace}(A)= \sum\limits _{i=1} ^{n}
a_{i,i}$
\end{center}
The trace is also a linear transformation from a square matrix to
the real numbers.
\textbf{Properties:}
\begin{enumerate}
\item $\operatorname{trace}(A+B) = \operatorname{trace}(A)+ \operatorname{trace}(B)$
\item $\operatorname{trace}(cA) = c\cdot \operatorname{trace}(A)$
\end{enumerate}
\textbf{Proof of Properties:}
\begin{enumerate}
\item Since the trace is a linear transformation therefore this property
is inherited. For the sake of proving that it is a linear
transformation. \\ $\operatorname{trace}(A+B)$\\ $= \sum\limits_{i=1}^{n}
(a_{i,i} + b_{i,i})$ property of matrix addition \\ $=\sum\limits
_{i=1} ^{n} a_{i,i} + \sum\limits _{i=1} ^{n} b_{i,i}$ property of
sums\\ $= \operatorname{trace}(A) + \operatorname{trace}(B)$\\
\item Like above the trace is a linear transformation therefore this property
is inherited. Proving that it is a linear
transformation. \\ $trace(cA)$\\ $= \sum\limits _{i=1} ^{n}
c\cdot a_{i,i}$ property of matrix scalar multiplication \\
$=c\cdot \sum\limits _{i=1} ^{n} a_{i,i}$ property of
sums\\ $= c\cdot \operatorname{trace}(A)$\\
\end{enumerate}
\textbf{Examples:} \\ Let $A = \begin{pmatrix}
2 & 4 & 6 \\
8 & 10 & 12 \\
14 & 16 & 18 \
\end{pmatrix}$ and $B = \begin{pmatrix}
9 & 8 & 7 \\
6 & 5 & 4 \\
3 & 2 & 1 \
\end{pmatrix}$ then:
\begin{itemize}
\item $trace(A+B)$\\ $= \operatorname{trace}(A) + \operatorname{trace}(B)$\\ $= (2 + 10 + 18) + (9 + 5 + 1)$\\ $= 45$
\item $trace(A)$\\ $= \operatorname{trace}(cA')$\\ $= c\cdot \operatorname{trace}(A')$ \\
$= 2\cdot \operatorname{trace}\begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \
\end{pmatrix}$\\ $= 2\cdot (1 + 5 + 9)$ \\ $=
30$
\end{itemize}
\textbf{Reference:}
\begin{itemize}
\item The Trace of a Square Matrix. Paul Ehrlich, [online] http://www.math.ufl.edu/~ehrlich/trace.html
\end{itemize} |
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