Wikipedia says that symmetric matrices are square ones, which have the property $A^T=A$. This assumes that one can have non-square $A^T=A$ and, because it does not satisfy the first property of symmetry, it is not symmetric. So, there can be non-symmetric $A^T=A$ matrices and the definition is right. Is it right? Or, definition is redundant and misleading, and it is better to define a symmetric matrix by single $A^T=A$ property and symmetric matrix squareness follows from symmetry rather than defines symmetry? The question in short: should I say that symmetric matrix is square, when define it?
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If you consider that if $A$ is a $n$-by-$m$ matrix, then $A^T$ is a $m$-by-$n$ matrix, so it follows that if $A$ = $A^T$ then a necessary condition is that $n=m$ as the matrices need to have the same number of rows/columns if they are equal, so $A$ must be a square matrix. |
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This is what Wiki says: "In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose". Now, it certainly follows from the definition that $\,A\,$ is symmetric if $\,A^t=A\,$, and from this it follows at once that $\,A\,$ has to be square, but mentioning this in the definition above can hardly be confusing: reduntant, yes. For example, you can check that many algebra books define a group as "a non-empty set with an binary operation...", and this "non-empty" thingy also is redudant as it follows at once from the axiom on the existence of a unity element...etc. Things like the above ones, besides being a little boring and futile to discuss a lot about, are designed mostly to avoid misunderstandings from beginner students. That's all, imfho. |
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No, if $A$ is an $n\times m$ matrix then $A^T$ is an $m\times n$ matrix, so if $A$ is not square, then $n\neq m$ and hence $A^T$ has different dimensions to $A$ and hence can't be compared with $A$. In that sense, $A^{T}=A$ implies $A$ square, but it also implies comparing matrices of different dimensions, which is something to avoid. (It's not technically wrong to say $A\neq B$ when $A,B$ are different dimensions, but it can cause confusion.) |
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