this post was submitted on 20 Sep 2023
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Explain Like I’m Five
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@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
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
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
Thanks for your explaination with examples. What is a column vector? Is it something like (1 2 3) which means move x upwards 1 unit, y up 2 units, z up 3 units?
@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.