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These are some of the concepts in maths, which I should know but sadly I keep forgeting them so here they are:

Integration

1. Integratio by parts: [Wiki]

Matrices

1. Useful identity involving the matrix inverse (from Bishop eq: C.5)
(P−1 + BTR−1B)−1BTR−1 = PBT(BPBT + R)−1
Proof
Consider above LHS : RHS
  • Multiply both side by R to the right. For LHS take R as R−1−1, and then use formula (AB)−1 = B−1A−1. This would give:
    (P−1 + BTR−1B)−1BT = PBT(R−1BPBT + I)−1
  • Now do the similar thing by multiplying P−1 to the left, to get:
    (I + BTR−1BP)−1BT = BT(R−1BPBT + I)−1
  • We have BT on both side, we can write that as BT−1−1, and then again apply (AB)−1 = B−1A−1 to get:
    (BT−1 + R−1BP)−1 = (R−1BP + BT−1)−1
  • Thus, LHS = RHS
Special case (Bishop C.6): (I + AB)−1A = A (I + BA)−1
2. Woodbury identity (these are usefula s depending on dimension it is cheaper to evaluate one side than the other) (Bishop C.7):
(A + BD−1C)−1 = A−1 - A−1B(D = CA−1B)−1CA−1
3. Cyclic property of trace operator for matrices (Bishop C.9)
Tr(ABC) = Tr(CAB) = Tr(BCA)