TWSB: We’re In the Matrix
The matrix of LINEAR ALGEBRA!!!
So someone asked the awesome dudes at Ask a Mathematician/Physicist why determinants of matrices are defined in the strange way they are. And one of the opening sentences of the Physicist’s response was:
“The determinant has a lot of tremendously useful properties, but it’s a weird operation. You start with a matrix, take one number from every column and multiply them together, then do that in every possible combination, and half of the time you subtract, and there doesn’t seem to be any rhyme or reason why.”
That needs to be a textbook definition somewhere.
Anyway. This was an especially interesting read for me, since we just learned about the role of the Jacobian matrix’s determinant when performing a change of variables for multiple integrals.
The Physicist has an excellent explanation of it (along with pictures!), but it basically comes down to the fact that the determinant of a 3 x 3 matrix, if we treat the columns of the matrix as vectors, is actually equal to the volume of the parallelepiped (coolest shape name or coolest shape name?) formed by the vectors. Think about if you have two vectors in the xy-plane. You can extend vectors out from each of the tails of the vectors so that you have a parallelogram like this:
Finding the determinant of the 2 x 2 matrix that describes those two vectors is the same as finding the area of the parallelogram formed by them. Add one more dimension and you get a parallelepiped for your shape and a volume for your determinant.
This has a buttload of applications—like I said, when performing a change of variables when doing multiple integration, but also for finding eigenvalues/eigenvectors and determining whether a set of vectors are linearly independent or not.
I was actually planning on making this a longer blog with an actual calculus application, but a) formatting that would take like 80 years for me and I’ve actually got to study sometime tonight and b) I’ve fallen into the “polar coordinates” article on Wiki and I don’t think I’ll be getting out anytime soon. THEY MAKE ME HAPPY, OKAY?!