rejected Appendix A
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A short draft appendix, apparently cut from Phil's notes on separation theory in curvilinear systems. It reviews inversion of X = AY using the cofactor formula [A^-1]pq = cof(Aqp)/det(A), noting the switched indices. It then treats X = B^T Y, defining the cofactor matrix C so that Y = [1/det(B)] C X.
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Appendix A: A small digression on inverting a matrix
From linear algebra, if we have a matrix equation in En
X = AY
we know that if detA ≠0 we can invert to get
Y = A-1Y
where
[A-1]pq = cof(Aqp)/det(A)
Notice that the subscripts are switched on the two sides, and
cof(Aqp) = (-1)p+q minor(Aqp)
where minor(Aqp) is the determinant of the (n-1)x(n-1) matrix obtained by crossing out row q and column p of the matrix A. Now suppose we are given this problem instead
X = BTY
where T means transpose. If det(BT) ≠ 0, we can invert to get
Y = (BT)-1X
where, noting that det(BT) = det(B),
(BT)-1pq = cof(BTqp)/det(BT)
or
(BT)-1pq = cof(Bpq)/det(B)
where now the indices are in the same order on both sides. To clarify notation, we might then define a matrix C where
Cpq ≡ cof(Bpq) " the cofactor matrix"
so we then have
(BT)-1pq = [1/det(B)]Cpq
so that our inversion is given by
Y = [1/det(B)] C X inversion of X = BTY