ways to write the determinant
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Phil's note dated 7.9.15 with later updates on 10/18/15 and 10.3.16, building on Appendix B of his Lagrange document. It rewrites det(M) as permutation sums with Parity and Levi-Civita symbols, derives a generalization where an arbitrary permutation c gives eps_c det(M) = sum of eps_a times products of matrix entries, and uses the rearrangement theorem. It shows this covers Sjamaar's formula on page 33 and records a form needed for tensors and wedge products.
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Ways to write a determinant PhL 7.9.15
Lagrange doc Appendix B has the key facts, and here I just do a little extension of those facts.
1. I show in (B.2.5) and following that you have these two ways to write det(M):
det(M) = Σa Parity(a) Π(z0,a; M) = Σa εa Mza = Σa εaa... a M1aM2a ...... Mna
det(M) = Σa Parity(a) Π(a,z0; M) = Σa εa Maz = Σa εaa... a Ma1Ma2 ...... Man (1)
2. Now here is a slight extension of these two results: start with the first line:
det(M) = Σa Parity(a) Π(z0,a; M)
= Σa Parity(a) Π(Cz0,Ca; M)
= Σa Parity(Ca) Parity(c) Π(Cz0,Ca; M) // Parity(Ca) Parity(c) = Parity(a)
= Σa Parity(a) Parity(c) Π(Cz0,a; M) // rearrangement
= Parity(c) Σa Parity(a) Π(c,a; M)
= Parity(c) Σa Parity(a) McaMca .... Mca c = Cz0
Had we started with the second form in 1, the a and c labels are reversed. So we get
Parity(c) det(M) = Σa Parity(a) McaMca .... Mca
Parity(c) det(M) = Σa Parity(a) MacMac .... Mac (2a)
In compact notation we have
εc det(M) = Σa εa Mca
εc det(M) = Σa εa Mac (2b)
In expanded notation
εcc... c det(M) = Σaa... a εaa... a McaMca .... Mca
εcc... c det(M) = Σaa... a εaa... a MacMac .... Mac (2c)
If you select c = z0 then (2c) replicates (1).
3. Now go back to this idea that
c = Cz0 = a permutation of (1,2,3...n).
ci = ΣjCij j since (z0)j = j
One can think of a permutation as a mapping
c: Zn → Zn Zn = {1,2,....n}
In this mapping we have
1 → c1 = c(1) ci = c(i) = ΣjCij j
2 → c2 = c(2)
...
i → ci = c(i)
....
n → cn = c(n)
4. Sjamaar writes on page 33
det(M) = ΣσSn sign(σ) M1,σ(1)M2,σ(2)......Mn,σ(n) Sn = set of all permutations of Zn
Well this is really the same as my result above where for fun I replace a by c
det(M) = Σc Parity(c) M1cM2c ...... Mnc c = σ
So I already have Sjamaar's bases covered!
**********************************************
5. Update on 10/18/15. Here is yet another form that I need in tensor and wedge
Consider the above result,
det(M) = ΣσSn sign(σ) M1,σ(1)M2,σ(2)......Mn,σ(n) Sn = set of all permutations of Zn
Write this as
det(M) = ΣP (-1)S(P) M1,P(1)M2,P(2)......Mn,P(n)
= Σaa... a εaa... a M1aM2a ...... Mna P{1,2....n} = {a1, a2, ...an}
We know the following from the rearrangement theorem: (Q can be on either side, a new fact!_
ΣP f(QP) = ΣPf(P) = ΣP f(PQ)
so can write the above as
det(M) = ΣP (-1)S(PQ) M1,PQ(1)M2,PQ(2)......Mn,PQ(n)
Now select a particular permutation Q = I which does this:
I{1,2....n} = {i1, i2, ....in } .
Then we have
det(M) = ΣP (-1)S(PI) M1,P(i)M2,P(i)......Mn,P(i)
= (-1)S(I) ΣP (-1)S(P) M1,P(i)M2,P(i)......Mn,P(i)
= εii... i ΣP (-1)S(P) M1,P(i)M2,P(i)......Mn,P(i)
which can be written
ΣP (-1)S(P) M1,P(i)M2,P(i)......Mn,P(i) = εii... idet(M)
Notice that this claim is valid only if {i1, i2, ....in } is a permutation of {1,2....n} .
Noted added 10.3.16. None of this "ways to write" doc relates to the cofactor method of appendix B.3. In fact only Section B.2 is involved, so maybe I can just edit that section to work in the main stuff above. Let's try that now in scratch. // Done, and added to Lagrange Appendix D.