Fan of #Vaporwave music
Non-technical, but enthusiastic about #NuclearEnergy and #FreeSoftware
#Vegan since 2019
Slowly self-studying advanced #math since finishing my undergrad in 2010

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  • 9 Comments
Joined 2 years ago
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Cake day: November 5th, 2022

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  • @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
    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
    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.