Stuad^Dib

@meowmeowmeow
Ah, I should have been more specific, but you pretty much have the right idea. A vector is, abstractly, something with a length and a direction, like a velocity or force in physics. But to actually make calculations with vectors it helps to represent them with lists of numbers like your example. The convention is that we write vectors vertically, hence “column vector.” Writing them horizontally as rows instead represents “covectors,” but I won’t get into the weeds on that.

@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

@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
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
4(a). “Orthonormal” combines “orthogonal” (sort of means the same as “perpendicular”) and “normal” (in this context means a vector with length 1). If a matrix is orthonormal, that means if we treat its columns as separate vectors, they’re all mutually perpendicular to each other and each have length 1. Why do we care enough to give this a special name? Well, it turns out orthonormal matrices rotate and reflect vectors, which has obvious uses to science and computer graphics.

@meowmeowmeow
3. Remember a matrix is like a function: multiply it with a column vector as input, and you get another column vector as output. In general, a matrix can transform vectors in all sorts of ways, but sometimes a matrix has special input vectors called “eigenvectors.” What makes them special is that, after multiplying, you get almost exactly the same vector you started with, but multiplied by some number called an “eigenvalue.” This page has some examples: https://www.mathsisfun.com/algebra/eigenvalue.html

@meowmeowmeow
2(b). The inverse is related to the identity. It’s sort of the “opposite” of a math object (a number, matrix, etc.) but in a specific way. When combining something with its inverse by some operation (like adding or multiplying) the result is the identity. For example: when adding, the inverse of x is -x because x+(-x) = 0. And when multiplying, the inverse of x is 1/x because x*1/x = 1. In the same way, when a matrix multiplies with its inverse, the result is the identity matrix.

@meowmeowmeow
2(a). In a lot of mathematical systems, the “identity” is the thing that “does nothing.” For example, when adding ordinary numbers the identity is 0 because adding 0 to any number does nothing - the other number stays the same. Similarly, when multiplying the identity is 1 because multiplying 1 with any number also does nothing. The identity matrix plays the same role - if you multiply any (square) matrix with the identity, you’ll get back the same matrix you started with.

@meowmeowmeow

1. A good way to think of matrices is as a kind of function. They take column vectors as "input” by multiplying with them, and the “output” from that product is another vector. The determinant measures how much a matrix stretches the space the input vectors come from. Big determinants stretch spaces way out, small ones shrink them way down, and negative ones reverse them like a mirror.