Practical uses of matrix multiplication

Solution 1:

A fundamental example is the multivariate chain rule. A basic principle in mathematics is that if a problem is hard, you should try to linearize it so that you can reduce as much of it as possible to linear algebra. Often this means replacing a function with a linear approximation (its Jacobian), and then composition of functions becomes multiplication of Jacobians. But of course there are many other ways to reduce a problem to linear algebra.

Solution 2:

Linear discrete dynamical systems, aka recurrence relations, are best studied in a matrix formulation $x_{n+1} = A x_n$. The solution of course is $x_n = A^n x_0$, but the point is to exploit the properties of $A$ to allow the computation of $A^n$ without performing all multiplications. As an example, take the Fibonacci numbers. The formula for them comes directly from this matrix formulation (plus diagonalization).

Don't forget the origins of matrix multiplication: linear change of coordinates. See, for instance, section 3.4 of Meyers's book (page 93) at http://web.archive.org/web/20110714050059/matrixanalysis.com/Chapter3.pdf.

See also http://en.wikipedia.org/wiki/Matrix_multiplication#Application_Example.