Question

I found this interesting relationship a couple minutes ago on my own, can anyone explain to me why they think this might be true?

Answer 1

\[\LARGE B^T=\frac{1}{\pi}\int\limits_0^{2\pi}[A_{jk}B_{kj}A_{jk}]dx\] So they don't have to be square matrices or anything, but I was trying to find a way to transpose a matrix by only doing matrix multiplications.

Here, B doesn't depend on x, but A does. A is just a matrix filled with sines starting at 1. For instance, it could look like this: \[\Large A_{23}= \left[\begin{matrix}\sin(x)& \sin(2x) & \sin(3x)\\ \sin(4x) & \sin(5x) & \sin(6x)\end{matrix}\right]\]

It doesn't matter where the sines are, they can be anywhere in any order as long as they're of different numbers.

Answer 2

I have no idea why this is true, but it seems to work every time.

Answer 3

i still dnt got ur observation can u eplain more?