Why is it called the symmetric part not form of a Matrix? I understand the math behind finding the symmetric part of a matrix being $1/2*(A+A')$ but it doesn't sound like the right use of English in my mind which makes me think I don't understand something.
Surely a part of a Matrix should refer to elements within it rather than turning the whole thing into a symmetric form by averaging it with its transpose. Saying the symmetric part makes me think of just the upper right and lower left elements. So When asked to find the maxmimum eigenvalue of the symmetric part I think of just those elements but clearly the eigenvalues depend on all elements of the matrix, hence my confusion over the terminology. I thought maths was precise? Hence I propose we change it to be the symmetric form of Matrix A :-).
 A: I agree that that there can be some misunderstanding. However, consider the following.
Consider a function $f(x)$. Then:
$$f(x) = f_o(x) + f_e(x),$$
where:
$$f_o(x) = \frac{f(x) - f(-x)}{2} ~\text{(it's its odd part)},$$
and
$$f_e(x) = \frac{f(x) + f(-x)}{2} ~\text{(it's its even part)}.$$
Specifically, $f_o(x)$ is an odd function and $f_e(x)$ is an even function, even if $f(x)$ is not odd, nor even, neither odd or even.
Here, even function stands for:
$$g(x) = g(-x),$$
while odd function stands for:
$$g(x) =-g(-x).$$
Similarly, for matrices we have that:
$$ A = A_{as} + A_{s},$$
where:
$$A_{as} = \frac{A - A'}{2} ~\text{(it's its anti-symmetric part)},$$
and
$$A_{s} = \frac{A + A'}{2} ~\text{(it's its symmetric part)}.$$
Specifically, $A_{as}$ is an anti-symmetric matrix and $A_s$ is a symmetric matrix, even if $A$ is not anti-symmetric, nor symmetric, neither anti-symmetric or symmetric.
Here, symmetric matrix means that:
$$B = B',$$
while anti-symmetric matrix stands for:
$$B = -B'.$$
