# Resources for matrices and its applications

I was preparing some presentation slides on basics of matrices and its application. Even though, many of the participants are familiar with basic matrix operation, I planned to explain them by starting from Matrix addition, Matrix subtraction, Matrix Multiplication and Matrix Inverse in comparison with real number operation.

My theme is learning a new concept = Comparison with known old concept + where the new concept differs from know old one + Why they differs.

In this process, I thought of comparing Real number operation with Matrix operation.

Though matrices are used in various algorithms for representing document and finding out it page rank,im looking for the example which is simpler and easy to understand.

In general, im looking for various resources on matrices and its application in real world which can be shown in creative way. Any book similar to Mathematical Nature walk for matrices, will be more useful.

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You can't talk about matrices without talking about linearity. It would practically be a sin to not mention their role as linear transforms on vector spaces, especially since that's probably the best motivation for the initially unintuitive form of matrix multiplication. A practical application from this angle would be video game graphics and geometry: understanding matrices puts every transformation (shifts, rotations, scaling, etc.) in a very nice conceptual framework that can be easily applied to otherwise difficult things, such as scaling an object mesh by calculating the new positions of all vertices in it. The fact that this is so intensely visual (it is geometry after all) is a very nice bonus that students are rarely granted in algebra, and shouldn't be passed up.

Another major application is the role of adjacency matrices in graph theory. If your students have never seen graph theory before then the use of these might not seem as immediate (after all, you'd have to explain the uses of graph theory itself), they are incredibly useful, and I would say that they're worth a mention at least. I believe Strang has a lecture as part of his Linear Algebra course where he uses these to solve circuitry problems, though I forget where the video is.

In terms of linking operations from old to new concepts: You probably won't have much trouble with addition/subtraction: for those operations a matrix can be treated as a grid of values. In my experience most students have no trouble with this, and only run into trouble with matrix multiplication. But if you introduce matrices as linear transformations, and matrix multiplication as matrix composition, then you can talk about the identity matrix, matrix multiplication, determinants, inverses, rank, and nullspaces in an extremely intuitive and easy to visualize manner. You can talk about what makes matrices matrices and not mere grids of numbers. I'd say this is your best bet for explaining "why they differ", since the why is all about linearity, which is motivated very well through geometric ideas.

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I'd show how to represent graphs with matrices - for example the adjacency matrix.

I'd mention that if $A$ is the adjacency matrix of a graph then $tr(A^3)/6$ is the number of triangles in $G$. This is the currently fastest algorithm for computing the number of triangles in a graph.

There are many others beautiful results relating linear algebra/matrix theory to graph theory. See also

http://mathoverflow.net/questions/96612/neat-results-from-algebraic-graph-theory

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