How should I define symmetric matrices?

Question

"A matrix A is a symmetric matrix if its transpose $A^T$ = $A$." This is my definition.

"A square matrix A is a symmetric matrix if its transpose $A^T$ = $A$." This is the textbook definition.

I understand that only square matrices can be symmetric, but I figured it would be all right to cut out the "square", as satisfying the equation $A^T$ = $A$ would ensure it is a square matrix.

So why can't I omit the "square"? (An error I was sternly reminded of.)

Any better definitions would also be greatly appreciated.

Answer 1

Given an $m\times n$ matrix $A=(a_{ij})$ and a $k\times l$ matrix $B=(b_{uv})$ with entries in a commutative ring $R$.

The matrices $A,B$ are equal if the mappings $f:[m]\times [n]\rightarrow R: (i,j)\mapsto a_{ij}$ and $g:[k]\times [l]\rightarrow R: (u,v)\mapsto b_{uv}$ are equal. Equality of maps means equality of the corresponding relations (sets).

Practically, this means that $A,B$ are equal if $A,B$ have the same size $(m=k,n=l)$ and for each pair $(i,j)\in[m]\times [n]$, $a_{ij}=b_{ij}$.

Well, if $A$ is an $m\times n$ matrix, then the transposed $A^T$ is an $n\times m$ matrix. By the above definition, the equality $A=A^T$ means that $m=n$ and $a_{ij}=a_{ji}$.

Answer 2

I think both definitions are perfectly valid (and mathematically equivalent), however I prefer the textbook definition.

The textbook definition makes a distinction between two cases. Consider the matrices $$A=\begin{bmatrix}1&1\\0&1\end{bmatrix}, B=\begin{bmatrix}1\\0\end{bmatrix}.$$

In some sense, the textbook definition says different things about them.

1. For $A$, the textbook definition tells you that $A$ is not a symmetric matrix, because $A^T\neq A$.
2. For $B$, the textbook definition tells you that $B$ is not a symmetric matrix because the concept of symmetric matrices is incompatible with non-square matrices.

I like this additional clarity in the definition. It makes it clear that $B$ is not symmetric because, in a sense, it doesn't even make sense to ask whether $B$ is symmetric.

Answer 3

You can say that an $m\times n$ matrix $A=(a_{ij})$ is symmetric if, and only if, $\forall i=1\ldots m, \forall j=1 \ldots n, a_{ij}=a_{ji}$, but you would encounter the problem of what to do if $a_{ji}$ is undefined. Accepting that $a_{ij} = \mathrm{undefined}$ is false, you could use that definition.

But anyway, your definition is OK. Trying to build something else is overcomplicating things!