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[This all goes in discussion of orthogonal operators]
(R_T of skew-sym gives interpretation of skew-sym: send everything to something perp)
[rotation is orthogonal skew-sym, and is a cute EG]
BTW, remember that Hermitian/Orthogonal means
=
...and since =
...that means T^*T = I
...which is better than ``T^*=T^{-1}''
...and much better than the horrible TT^*
(used by illiterates)
Notably, some *left* invertible operators are Hermitian,
as in the shift (a,b,c,...) -> (0,a,b,c,...)
on l^2
(it does preserve distance!)
...or at least you need a more careful definition
(norm-preserving -> injective, and in finite dimension, this means iso;
also useful for discussing isometric embeddings of one space into another;
there's also "coisometric", or rather orthogonal projection)
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Involutions in physics
The CPT theorem states that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.
Connections with physics
CPT
T = reverse time
P = parity (multiply space by -I as well; note that this flips orientation)
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can you define the Jordan structure on quadratic forms
more intrinsically?
[not really: need to do it w/r/t a non-degenerate one;
can sorta get a torsor structure:
norm a / norm b = norm c / norm d]
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STUFF
module theory
(recall simple, irreducible, semisimple, totally reducible;
this module theory is the algebra underlying this,
and classification of operators really is just *models* for these.
EG of 1 1
0 1
...as irreducible but not simple)
[fields are semisimple;
Z and K[x] are *not*]
Not operator is cyclic:
a *cyclic* K[T] module is K[T]/p(T),
where p(T) is the characteristic polynomial,
but EG, if T=Id_2, the module is K[T]/(T-1) + K[T]/(T-1)
lineary algebraically, it's "has a cyclic vector"
(and is called "regular")
In generally the module is *not* K[T]/p(T)
BTW, call Z/q = Z/p^r a *primary* group
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classic quotient functions:
mod n
trig: cos, sin
(periodic so on circle)
Rayleigh quotient
(descends to quotient)
[map between vector spaces can be projectivized]
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``(over the complexes)
An algebra of operators is abelian
iff it consists only of normal operators''
A commutative W^*-algebra is cyclic (generated by functions in a normal operator A)
- If it's an R-algebra, A is hermitian
...and thus any commuting set of normals is functions in a single normal
- if it's a C-algebra, A is non-degenerate skew-hermitian
...and thus any commuting set of hermitians is functions in a single hermitian
A and B hermitian, and A + iB normal -> AB = BA
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\begin{thm}[Quaternionic spectral theorem]
I don't know the statement of this,
though normal operators are apparently orthonormally diagonalizable
(by same proof as in complex case).
``An operator $T$ on a quaternionic inner product space
is orthonormally diagonalizable
iff ???.''
\end{thm}
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Quaternionic notes:
- physicists have been most interested in quaternions
- Teichm\"uller proved spectral theorem for normal operators over quaternions
in his thesis?
- only an algebra over reals (b/c C not in center)
- every normal operator is unitarily equivalent to it adjoint!
- quaternionic multiple of eigenvector has different eigenvalue!
(b/c of non-commutativity)
[can choose a unique representative if you like]
- forgetting vector space over H -> vs over C (via the embedded C -> H)
is called ``the symplectic image of H''
Heuristically,
``normal operators behave like complex numbers''
(duh: they're multiplication by complex numbers along some axes)
Recall:
every quaternion is conjugate to q^*
(they are determined by norm and real part)
...similarly, every normal operator on quaternionic space is conjugate to q^*
(centralizer of centralizer of a real is the reals)
[duh: center is reals: centralizer of center is whole group,
centralizer of whole group is center]