Other operations

POWER: Given an $ n\times n$ matrix (square matrix) $ A$, we can define the $ k$-th power $ A^k$ of $ A$ by

$\displaystyle A^k = \underbrace{A \cdots A}_{k} .
$

TRANSPOSE: Given an $ m\times n$ matrix $ A$, we can define the transpose $ A^T$ of $ A$. In particular, $ (A^T)^T = A$. For example,

$\displaystyle \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_...
...{bmatrix}
a_{11} & a_{21} \\
a_{12} & a_{22} \\
a_{13} & a_{23}
\end{bmatrix}$

PROPERTIES OF . If $ A + B$ and $ A B$ are appropriately defined, we have $ (A + B)^T = A^T + B^T$ and $ (AB)^T = B^T A^T$.



Department of Mathematics
Last modified: 2005-10-21