Matrix Multiplication

Re:Matrix Multiplication by bandrami on Friday November 25, 2016 @09:36AM (#13036961)

Bingo. We did matrices in high school and even through a first linear algebra course in undergrad I got more or less competent at the mechanics of matrix multiplication without ever having any sense of why anyone would want to do this or why these rules, as opposed to some other set of rules that were equally formally valid, were of any interest. I could even tell you what the eigenvectors of a matrix were but had no sense of what that actually meant in terms of a linear transformation that matrix represented. Linear algebra is probably the area of math that is both universally important and completely skimped on by our education system.

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Re:Matrix Multiplication by t3nary on Friday November 25, 2016 @09:43AM (#13037000)

I feel very similar. Do you have any recommendations for resources to learn about why matrix multiplication is defined that way?

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Re:Matrix Multiplication by bandrami on Friday November 25, 2016 @10:15AM (#13037253)

"All the mathematics you missed but need to know for graduate school"[1] helped me a lot (and, in fact, I had and did).

Once I had finished that, Cullen's "Matrices and linear transformations"[2] was really helpful too. But I wouldn't do Cullen if you're still, as I was, floundering with the concepts of why you're doing this in the first place. It's great once you have those concepts down.

[1]: https://www.amazon.com/All-Mathematics-You-Missed-Graduate/d...

[2]: http://store.doverpublications.com/0486663280.html

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Re:Matrix Multiplication by fsloth on Friday November 25, 2016 @10:26AM (#13037341)

Garrity's book looks exactly what I've been looking for for brushing up on basic math skills after a decade of fairly low-brow work after my MSc. Thanks for the reference!

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Re:Matrix Multiplication by bandrami on Friday November 25, 2016 @10:46AM (#13037466)

Hope it helps! It definitely got me through the first year of grad school...

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Re:Matrix Multiplication by cousin_it on Friday November 25, 2016 @10:16AM (#13037262)

Let's say you have a system with N possible states, evolving in discrete steps. At each step, the system has some probability of switching from any state to any other state. That gives you an NxN matrix of switching probabilities. For example, if the system always stays in the same state as it started, the switching probabilities are an identity matrix (1 on the diagonal and 0 everywhere else).

Now let's see what happens after two steps. If the system started out in state i, what's the probability that after two steps it will end up in state k? Well, it's the sum over all possible paths. In other words, the sum of probabilities of i->j->k for all possible j. In other words, the sum of p_{ij} times p_{jk} for j from 1 to N. But that's exactly the definition of multiplying a matrix by itself.

Now it should be easy to understand that whenever you have matrices that represent transformations of some object, composing transformations will correspond to multiplying matrices.

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Re:Matrix Multiplication by darrickw on Friday November 25, 2016 @09:49AM (#13037046)

Check out this series of videos "The Essence of Linear Algebra"[1] for a really powerful visual and intuitive explanation. It starts with vectors and builds to matrix multiplication and further to several other topics.

[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

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Re:Matrix Multiplication by ww2 on Friday November 25, 2016 @10:46AM (#13037471)

If you compose two linear transformations into one step, and apply distributive property, you get the matrix multiplication rule.

http://math.stackexchange.com/questions/31725/intuition-behi...

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Re:Matrix Multiplication by jperras on Friday November 25, 2016 @10:55AM (#13037546)

Axler's book, "Linear Algebra Done Right", is widely considered to be one of the best texts on linear algebra that focuses less on the mechanics and more on the structure of linear operators on vector spaces.

http://linear.axler.net/

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Re:Matrix Multiplication by CalChris on Friday November 25, 2016 @12:08PM (#13038124)

Yes, and the abridged version (no proofs, examples, and exercises) is downloadable for free. He avoids determinants altogether (well, until the 10th chapter).

http://linear.axler.net/LinearAbridged.html

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