We describe one more common operation we can perform on matrices. Informally, to transpose a matrix is to build a new matrix by swapping its rows and columns.
Given an $m\times n$ matrix $A$, its transpose is the $n\times m$ matrix $\transpose{A}$ given by \begin{equation*} \matrixentry{\transpose{A}}{ij}=\matrixentry{A}{ji},\quad 1\leq i\leq n,\,1\leq j\leq m. \end{equation*}
It will sometimes happen that a matrix is equal to its transpose. In this case, we will call a matrix symmetric. These matrices occur naturally in certain situations, and also have some nice properties, so it is worth stating the definition carefully. Informally a matrix is symmetric if we can “flip” it about the main diagonal (upper-left corner, running down to the lower-right corner) and have it look unchanged.
The matrix $A$ is symmetric if $A=\transpose{A}$.
You might have noticed that Definition SYM did not specify the size of the matrix $A$, as has been our custom. That is because it was not necessary. An alternative would have been to state the definition just for square matrices, but this is the substance of the next proof.
Before reading the next proof, we want to offer you some advice about how to become more proficient at constructing proofs. Perhaps you can apply this advice to the next theorem. Have a peek at Proof Technique P now.
Suppose that $A$ is a symmetric matrix. Then $A$ is square.
We finish this section with three easy theorems, but they illustrate the interplay of our three new operations, our new notation, and the techniques used to prove matrix equalities.
Suppose that $A$ and $B$ are $m\times n$ matrices. Then $\transpose{(A+B)}=\transpose{A}+\transpose{B}$.
Suppose that $\alpha\in\complex{\null}$ and $A$ is an $m\times n$ matrix. Then $\transpose{(\alpha A)}=\alpha\transpose{A}$.
Suppose that $A$ is an $m\times n$ matrix. Then $\transpose{\left(\transpose{A}\right)}=A$.