Generalized pythagoras theorem
For me, the easiest way to remember a definition is to keep in mind a typical illustration of the same. For example, Euclid's algorithm gives me all the properties I need to remember about algorithms, including the finite termination requirement.
When it comes to theorems, I consider Pythagoras theorem as a perfect illustration. Each one of the many proofs available today exemplifies the myriad ways in which a mind reaches its rational conclusions and it is no exaggeration to say that mathematical logic has its origins in geometrical proofs.
It is easy to imagine the 3-dimensional version of the Pythagoras theorem: The square of the area of an inclined triangle resting at the corner of a room is equal to the sum of the squares of the areas of the projections of the triangle on the two walls and the floor. But can we think of higher dimensional theorems, and if we can, how are we going to prove them?
It turns out that the theory of compound matrices provides a way to state and prove them. See
Generalized Pythagoras Theorem
Sunday, November 11, 2007
