transpose Transpose 2001-11-20 23:49:15 2005-03-18 22:11:39 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} The \emph{transpose} of a matrix $A$ is the matrix formed by ``flipping'' $A$ about the diagonal line from the upper left corner. It is usually denoted $A^t$, although sometimes it is written as $A^T$ or $A'$. So if $A$ is an $m \times n$ matrix and $$ A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$ then $$ A^t = \begin{pmatrix} a_{11} & a_{21} & \cdots & a_{m1} \\ a_{12} & a_{22} & \cdots & a_{m2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{nm} \end{pmatrix} $$ Note that the transpose of an $m \times n$ matrix is a $n \times m$ matrix. \subsubsection*{Properties} Let $A$ and $B$ be $m \times m$ matrices, $C$ and $D$ be $m\times n$ matrices, $E$ be an $n\times k$ matrix, and $c$ be a constant. Let $x$ and $y$ be column vectors with $n$ rows. Then \begin{enumerate} \item $(C^t)^t = C$ \item $(C+D)^t = C^t + D^t$ \item $(cD)^t = cD^t$ \item $(DE)^t=E^tD^t.$ \item $(AB)^t = B^t A^t.$ \item If $A$ is invertible , then $(A^t)^{-1} = (A^{-1})^t $ \item If $A$ is real, $\operatorname{trace}(A^tA) \ge 0$ (where $\operatorname{trace}$ is the trace of a matrix). \item The transpose is a linear mapping from the vector space of matrices to itself. That is, $(\alpha A + \beta B)^t = \alpha (A)^t + \beta (B)^t$, for same-sized matrices $A$ and $B$ and scalars $\alpha$ and $\beta$. \end{enumerate} The familiar vector dot product can also be defined using the matrix transpose. If $x$ and $y$ are column vectors with $n$ rows each, $$ x^t y = x \cdot y $$ which implies $$ x^t x = x \cdot x = ||x||_2^2 $$ which is another way of defining the square of the vector Euclidean norm.