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Mathematik-Online problems:

Problem 24: The Trace of Matrices

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
The trace $ {\operatorname{Tr}}(A)$ of an $ n\times n$-matrix is defined as sum over all elements on the diagonal, i.e. $ {\operatorname{Tr}}(A):=a_{11}+ \ldots + a_{nn}$.
a)
Find $ {\operatorname{Tr}}(AB)$ and $ {\operatorname{Tr}}(BA)$ for the matrices

$\displaystyle A=\left(\begin{array}{rr} 1 & -2 \\ 3 & 4 \end{array} \right) \qu... ...mbox{und}} \quad B=\left(\begin{array}{rr} 2 & 3 \\ 1 & -1 \end{array}\right). $

b)
Show that $ {\operatorname{Tr}}(CD)={\operatorname{Tr}}(DC)$ holds for arbitrary $ n\times n$-matrices $ C$ and $ D$.
c)
Is $ {\operatorname{Tr}}(CD) = {\operatorname{Tr}}(C)\cdot {\operatorname{Tr}}(D)$ true for all $ C, D\in\mathbb{R}^{\mathit{n\times n}}$ ?

(Authors: Strauss/Höfert)

Solution:

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  automatisch erstellt am 14. 10. 2004