Symmetric matrices
(written elsewhere)
- spectral theorem explains them as operators
- can also be interepreted as quadratic forms
ultimately, they're all diagonal in disguise
nice easy EG:
a b
b a
BTW, a product of symmetric matrices need *not* be symmetric
EG
2 * 0 1 = 0 2
1 1 0 1 0
...but it is if they commute
(which means they preserve the same basis, etc.)
(likewise not Lie algebra)
...and this is why they're weird as a *set*:
(from the *operator* POV)
their algebraic properties are weird
OTOH, they are a Jordan algebra!
(...and I have no idea how to interpret this)
You can take sum (which is nice from quad form POV)
*philosophical lesson*:
properties of a collection
instead of just properties of each representative