Go to content
If a 2x2 matrix M has eigenvalues \lambda_1 and <\lambda_2 and corresponding eigenvectors \left(\begin{array} & s_1 \\ t_1 \end{array} \right) and \left(\begin{array} & s_2 \\ t_2 \end{array} \right),
then M can be expressed in the form S\Lambda S^{-1}, where S = \left(\begin{array} & s_1 & s_2 \\ t_1 & t_2 \end{array} \right) and \Lambda = \left(\begin{array}{rr} \lambda_1 & 0 \\ 0 & \lambda_2 \end{array} \right) .
This is called reducing M to diagonal form.
Change
 Steps  
All
Change
 Steps  
All
O
O
O
O
    
+
r
-
    

Summary/Background

Arthur CayleyArthur Cayley (1821 1895) was a British mathematician. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years. He published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work is in developing the algebra of matrices, work in non-euclidean geometry and n-dimensional geometry.
He was consequently able to prove the Cayley-Hamilton theorem -- that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way -- as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups.
In 1925, it was discovered that matrices and complex numbers were a necessary tool in describing the behaviour of atomic systems.

Software/Applets used on this page

jsMath
This page uses jsMath You can get a better display of the maths by downloading special TeX fonts from jsMath. In the meantime, we will do the best we can with the fonts you have, but it may not be pretty and some equations may not be rendered correctly.

Glossary

cayley-hamilton theorem

Every square matrix satisfies its own characteristic equation.

group

a set of elements together with a binary operation which is closed and associative, for which an identity exists in the set, and for which every element has an inverse in the set.

matrix

a rectangular or square grid of numbers.

permutation

a selection of r objects from a set of n unlike objects where the order of selection is important

work

Equal to F x s, where F is the force in Newtons and s is the distance travelled and is measured in Joules.

Full Glossary List