this post was submitted on 20 Sep 2023
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Explain Like I’m Five
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@meowmeowmeow
4(b). An equivalent property of an orthonormal matrix is that its transpose (flipping a matrix so that every row becomes a column and every column a row) is equal to its inverse. Unitary matrices are almost exactly the same, except that they use complex numbers instead of just real ones, and instead of taking the transpose to get the inverse you also have to take the complex conjugate of every element. There's a lot more to them, but this is the best way I can keep it ELI5.
@meowmeowmeow
5. A diagonal matrix is what it sounds like - all of the (nonzero) entries are on the diagonal, from the top left corner to the bottom right. Why do we care? All sorts of calculations are easier with diagonal matrices, which is great for lazy mathematicians and efficient programmers. Some matrices aren't diagonal, but "diagonalizable," meaning we can shuffle them around into a similar diagonal matrix by using their eigenvectors, which comes in quite handy.
@meowmeowmeow
6. I assume by this you mean "how to compute the product of two matrices?" If you mean something else, let me know. Basically, if we want to multiply two matrices A and B to get their product AB, we multiply every row of the matrix on the left with every column of the matrix on the right. I can't really typeset matrices here, so hopefully the examples on this page are helpful:
https://www.mathsisfun.com/algebra/matrix-multiplying.html