I very much like the form of this, yet I wish that the dependencies were ordered differently, such that one would not need to jump around (e.g. eigenvalues before determinants, since the latter are defined by the former).
Worth checking the Wikipedia page on matrix calculus to see which version of the notation you use! This is in numerator (iirc) and so d (vec x) / d(t) = (vec x). I was used to d (vec x) / d(t) = transpose(vec x), and so this cookbook confused me for ages :S
When I learned about matrices in school it really bored me, even though math in general really captures my interest. Can anyone recommend a good source that makes matrices interesting and captivating?
Say you start with a small system of equations and you find out how you can solve them. Then you solve a bunch more, but then you start to notice that you don't like writing out the x, y, and z's, so you just keep the coefficients, in a box of sorts. Then you translate what you were doing into operations on the rows of this ... matrix you call it. After solving some more equations, you start to forget where these things come from. Surely they come from equations, but because the coefficients, plus signs and equal signs disappear, you are left with a box of numbers really.
What happens next is that different questions start to come to mind. The matrix attains a life of its own. You start to think questions like "Well, we have some x, y, z's and this box of numbers somehow transforms them into the right hand side's a, b, c's". Ok, the matrix is some kind of "transformation" now. You look at its properties. It appears that aX + bY can be computed entry-wise, so maybe this is a fundamental property and you call it linearity. Hmm. At some point you forget about the boxes of numbers and now you have bonafide transformations. From what? To where? You call these...vector spaces. And so on.
Further down the line you become preoccupied by questions of a different flavour, like what happens if you want something like Ax = cx. That is, we get x back with minimal distortion. Fairly simple question that leads to a lot of math.
It also just happens that when you consider mathematical functions of several dimensions, and matrices become apparent when you consider linear approximations. You've built this great theory for them and now that theory seems to really help in these kinds of problems. That is, a LOT of computational problems in this world.
The expressiveness and power you get from being able to cast and solve problems in matrix form is similar to the lightbulb that pops up when one “gets” the relational algebraic power underpinning SQL.
What is this? These pages are a collection of facts (identities, approximations, inequalities, relations, ...) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference.
In case you were, uh, expecting to see Keanu Reeves in a kitchen apron.
Thanks. I was just now straining to remember what food was shown in the films. A bowl of snot. A juicy & delicious steak. The oracle's cookies. I actually never realized before that food is such a theme. Makes sense, it is a very non-machine activity to consume food.
I only ever took one course in university on linear algebra and numerical methods, and I'm an electronic engineering student. But my hobbies (mathematical classical and Marxian economics) necessitate a deep understanding of matrices, sets and vector algebra. I hope this will help me!
Dang it! I thought this was gonna be a cookbook from the movie Matrix. Like how to bake the cookies the oracle made, or the steak Cypher ate. That looked good.
I often say: "If Neo had simply read Linear Algebra textbooks instead of hacking around all night long, we would have known that the matrix is a representation of a linear map (morphism of vector spaces) with respect to given bases in the domain and codomain.".
Mathematics lets you see and understand the matrix.
In a large pot over medium heat, melt butter. Cook onion and celery in butter until just tender, 5 minutes.
Pour in chicken and vegetable broths and stir in chicken, noodles, carrots, basil, oregano, salt and pepper.
Bring to a boil, then reduce heat and simmer 20 minutes before serving.
There is no spoon.