Under what non-trivial circumstances does (A+B)^2=A^2+B^2?
How is this connected with Positron Emission Tomography?
How is this connected with Positron Emission Tomography?
3 Answers
Under the circumstance that
Explanation:
We want to find when
We start by expanding the left hand side using the perfect square formula
So we see that
See below.
Explanation:
If
then necessarily
Some possibilities...
Explanation:
Given:
(A+B)^2 = A^2+B^2
A couple of possibilities...
Matrices
If
For example:
A = ((0, -1, 0, 0), (1, 0, 0, 0), (0, 0, 0, -1), (0, 0, 1, 0))
B = ((0, 0, -1, 0), (0, 0, 0, 1), (1, 0, 0, 0), (0, -1, 0, 0))
Then:
AB = ((0, 0, 0, -1), (0, 0, -1, 0), (0, 1, 0, 0), (1, 0, 0, 0))
BA = ((0, 0, 0, 1), (0, 0, 1, 0), (0, -1, 0, 0), (-1, 0, 0, 0))
Field of characteristic
In a field of characteristic
So:
(A+B)^2 = A^2+color(red)(cancel(color(black)(2AB)))+B^2 = A^2+B^2