He recommended “A Course in Number Theory and Cryptography” by N. Koblitz, QA 241.K672. Carleton library has two copies, so next time I am on campus, I shall sign one out. I guess we’ll find out how much of a dummy with limited algebra skills I am.

It’s all about copious spare time.

Setup: If A is self-adjoint, and W is an A-invariant subspace ⇒ W^⊥ is A-invariant.

Want: ∀ x∈W^⊥, Ax ∈ W^⊥〈Ax,w〉 = 0 ∀ w ∈ W ⇐ Orthonormal

Given:〈Ax,w) =〈x,Aw〉 ⇐ Self-Adjoint

Aw∈W ⇐ W is A-invariant

then〈x∈W^⊥, Aw∈W〉= 0

Since any self-adjoint matrix is ortho-diagonalizable, if A is self-adjoint, then ∃ an orthonormal basis B∈ℂⁿ made out of eigenvectors such that [A]_B

Want: k=n (that is, an orthonormal basis made out of eigenvectors).

Proof by contradiction: Suppose k less then n

Given:

W=span(v₁, v₂, .. v_k), A-invariant (this is trivial, but see Appendix A)

then W⊥ is A-invariant, then A is restricted to subspace W^⊥

A_W^⊥: W^⊥ → W^⊥

Then ∃ v∈W^⊥, an eigenvector of A_W^⊥.

But since A_W^⊥v = A_v, v is an eignevector to of A perpendicular to W.

We assumed that S is maximal, but we ended up with a contradiction, since the set {v₁, v₂, .. ,v_k, v/ ||v||} is an orthogonal set of eigenvectors.

So k must be equal to n. As a result, A is orthodiagonalizable.

Conclusion

HTML is really not suited for doing math.

Appendix A

If λ₁≠λ₂,〈v₁,v₂〉must be 0

Here is how: Av₁=λ₁v₁, Av₂=λ₂v₂

Given that: λ₁〈v₁,v₂〉 = 〈λ₁v₁,v₂〉= 〈Av₁,v₂〉= 〈v₁,Av₂〉= 〈v₁,λ₂v₂〉=λ₂〈v₁,v₂〉⇒ λ₁〈v₁,v₂〉= λ₂〈v₁,v₂〉

Since λ₁≠λ₂, 〈v₁,v₂〉=0

.

QED

For an element b ∈ R consider the subset bR/aR = {r + Ra : r ∈ bR} of R/aR.

Let r + Ra, s + Ra ∈ bR/aR so that r, s ∈ bR, then r + s ∈ bR which gives (r + Ra) + (s + Ra) = (r + s) + Ra ∈ bR/aR.

Let r ∈ Ra ∈ R/aR and s + Ra ∈ bR/aR so that s ∈ bR, then rs ∈ bR implies that (r + Ra) · (s + Ra) = (rs) + Ra ∈ bR/aR.

Hence bR/aR is an ideal in R/aR

Just in case you weren’t sure that it was.

It’s almost 2 am, what do you expect?

Oh, and I goofed while writing a game theory midterm last week, and only got 92/100.

I forgot to do part II of question 1, inspite of the note on the test paper saying “Don’t forget to do part (ii) below”.

Argh!

Oh, and should I do a blurb on the math behind error correction, specifically, as it’s implemented on CDs?

You want to divide two thirds by five sixths. So you use GNU bc

stany@Gilva:~[11:13 PM]$ echo  "(2/3) / (5/6)" | bc -l
.79999999999999999999
stany@Gilva:~[11:14 PM]$

Matlab, which is arguably better at math then bc is, gives me this:

EDU>> (2/3) / (5/6)

ans =

    0.8000

EDU>>

2*6 / 3*5 = 12/15 = 4/5 = 0.8 so Matlab is correct, and bc is wrong.

Now, why did I discover this thing after I wrote my final exam? So that I could learn how to do it by hand, that’s why. It would have rocked my world about 3 months ago, but I probably would have failed the exams, relaying on it to do everything for me.

Oh, and notation it uses for CFG/PDA is somewhat different from one used in Sipser (Which is a damn fine book, BTW. Unlike Lawson it actually covers CFLs.).

P.S Mark Lawson actually makes reasonably good notes available on his page about basic automata. It still doesn’t go into CFLs/PDAs.