Posted by: danb | December 6, 2008

Cayley and Hamilton

We are now studying the Cayley Hamilton theorem, which states that a square matrix satisfies its characteristic equation.  But, if someone asked you who Cayley and Hamilton were, could you tell them?  I couldn’t…before writing this.

 

William Rowan Hamilton was an Irish mathematician born in 1805.  At a very young age his tongue for foreign languages was noted, and he later attended Trinity College in Dublin, where he studied physics.  Before he graduated, he had become a professor of astronomy.  However, because he was so talented in math and science in general, the Trinity administration allowed him to pursue studies in anything that would benefit science as a whole.  His studies included advancements in optics, dynamics, linear algebra, classic mechanics and quaternions.  His discovery of quaternions is amongst his most notable achievements.  A quaternion is an extension of the normal 2-dimensional complex plane into 4 dimensions.  His book, Elements of Quaternions, was published posthumously.  The applications of quaternions extend to rotations (I’d be curious to see how quaternions apply to our class, so if anyone is experienced with this, let me know).  Another topic of Hamilton’s research was the Hamiltonian system, which, for those of you lucky enough to have taken the class, was studied in Math 106. This system of differential equations shows the sum of potential and kinetic energy for a closed system.  However, the application can be extended to other fields.  Here is a quick review of the form:

\dot p = -\frac{\partial \mathcal{H}}{\partial q}
\dot q =~~\frac{\partial \mathcal{H}}{\partial p}

Hamilton died in 1865 though his impact on science and mathematics remains.  He has a mathematical institute named in his memory, the Hamilton Institute.

Arthur Cayley was a British mathematician born in 1821, and similarly to Hamilton, was skilled in foreign languages at a young age.  Later, Cayley, coincidentally, attended Trinity College in Cambridge.  During his undergraduate years, he published several papers on various fields in math, including responses to works by Lagrange and Laplace.  However, Cayley chose to practice law after college.  While a lawyer, Cayley went to a presentation by Hamilton on quaternions.  This was the first contact between the two men; Cayley and Hamilton together proved their theorem, that any square matrix is a root of its characteristic equation.  Cayley has many other mathematical concepts to his name, including Cayley’s theorem, Cayley-Dickson construction, and the Cayley graph.  Cayley’s theorem says that every group G is isomorphic to a subgroup of the symmetric group on G.  For example, Z3={0,1,2} is isomorphic to {e, (123), (132)}.  Cayley Dickson construction refers to the sequence of algebra over a field of real numbers.  Each field in the sequence has double the dimension of the previous.  Just as Hamilton described quaternions, Cayley Dickson construction allows one to go from complex numbers to quaternions to octonions to sedions.  The Cayley graph is a graph that shows the structure of a group, with a set of generators.  It is easy to see with this graph of D_4 with generators alpha and beta.  When you multiply the identity element by alpha, it takes you to alpha.  Multiplying this by beta takes you to alpha*beta. 

 

Together, these two mathematicians contributed a considerable amount to the topics we have been studying.  Unfortunately, I did not go into great depth into any one of the works of Cayley or Hamilton, but I think that this would serve as a good project for a future blog post if anyone is interested.

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Responses

  1. The picture that didn’t upload on that post can be found at this link: http://upload.wikimedia.org/wikipedia/commons/5/58/Cayley_Graph_of_Dihedral_Group_D4.svg.

    Sorry about that!

  2. Dan, do you have any more information about quaternions? I know you were staying in this past Friday to read up on them, and I just wanted to know what you discovered…


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