Also known as "inner products", according to a song of the same name (whose lyrics are reproduced below).
Anyway, the song is by Peter Dong, and it was part of "Les Phys" - a musical he wrote to fulfill his BA thesis requirement for Physics and Music, at Harvard.
(the libretto can be found at http://schwinger.harvard.edu/~georgi/LesPhys.PDF)
-- BEGIN LYRICS --
Don?t ask me what the reason was I took Math 55.
I wonder now if anyone does get out of it alive.
As soon as I get into class, I?m fighting off a swarm
Of positive-definite non-degenerate symmetric bilinear forms.
Oh! You stay up all night Tuesday working on them in your dorm.
You get the same topology however you transform.
You put ?em back together and you get your favorite norm.
Those positive-definite non-degenerate symmetric bilinear forms.
My roommate?s in Math 22, and his TFs instruct
That such a form it?s okay to call an ?inner product.?
But powers that be declared that ?55-ers must conform
To ?positive-definite non-degenerate symmetric bilinear forms.??
STEVE and ALBERT:
Oh! You stay up all night Tuesday working on them in your dorm.
You get the same topology however you transform.
You put ?em back together and you get your favorite norm.
Those positive-definite non-degenerate symmetric bilinear forms.
STEVE:
Though I might stay up all night?and believe me, yes I do?
Compared to him I just might be here finding two plus two.
But I won?t ever have to work?and this thought keeps me warm?
With positive-definite non-degenerate symmetric bilinear forms.
STEVE and ALBERT:
Oh! You stay up all night Tuesday working on them in your dorm.
You get the same topology however you transform.
You put ?em back together and you get your favorite norm.
Those positive-definite non-degenerate symmetric bilinear forms.
((the next two stanazas, sung by Steve and Albert respectively, are to be sung simultaneously))
STEVE:
Let?s assume orthonormal bases,
Take the magnitude of the two.
Find the square and radical,
Assign this morphism. Then we?ll do
Some checks to see if these commute,
And yes, they?re quite convivial.
And is this in the vector space R1?
Of course! It?s trivial!
In matrix form, Euclidean scalar
Product must be true.
So let?s assign a zeta such that
Zeta equals two.
With all these transformations,
Eigenvalues, we can see
That, ipso facto, two plus two is four.
Q.E.D.
ALBERT:
Let V be defined as real,
And define a metric on PV;
We found that alpha must satisfy
The triangle inequality.
Now show the distance function
In this space that?s projectivial,
Must be complete and bounded, so
PV?s compact. It?s trivial!
Restrict this to R3
And use high school stereometry.
This reduces to elementary
Spherical geometry.
The sphere of dimension n?1,
Modulo antipodality,
Is homeomorphic to the projective space.
Q.E.D.
STEVE and ALBERT:
Those positive-definite non-degenerate symmetric bilinear forms.
Those positive-definite non-degenerate symmetric bilinear forms!
Oh! You stay up all night Tuesday working on them in your dorm.
You get the same topology however you transform.
You put ?em back together and you get your favorite norm.
Those positive-definite non-degenerate symmetric bilinear forms.
Those positive-definite non-degenerate symmetric bilinear forms!
They?re positive and definite!
They?ll never be degenerate!
ALBERT: Unless we confine them to an isotropic subspace. Then they would be degenerate. Oh,
wait, I said they were positive-definite, didn?t I?
STEVE: Uh?I guess.
STEVE and ALBERT:
Those positive-definite non-degenerate symmetric bilinear forms!
Those positive-definite non-degenerate symmetric bilinear forms!
-- END LYRICS --
To the best of my knowledge, none of this is copyrighted.