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

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**.## Summary/Background

Arthur 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.

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

## 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

### polynomial

an expression which has whole-numbered powers of the variable, x, multiplied by coefficients, and added together.

Examples: y = 2x, y = 1+xÂ²,y = 3-2x+7xÂ³

Examples: y = 2x, y = 1+xÂ²,

### union

The union of two sets A and B is the set containing all the elements of A and B.

### work

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

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AQA A-Level (UK - Pre-2017) | FP4 | Matrix algebra | Cayley-Hamilton theorem | - |

AQA A2 Further Maths 2017 | Pure Maths | Further Matrices | Cayley-Hamilton Theorem - Extra | - |

AQA AS/A2 Further Maths 2017 | Pure Maths | Further Matrices | Cayley-Hamilton Theorem - Extra | - |

CCEA A-Level (NI) | FP1 | Matrix algebra | Cayley-Hamilton theorem | - |

Edexcel A-Level (UK - Pre-2017) | FP3 | Matrix algebra | Cayley-Hamilton theorem | - |

Edexcel AS Further Maths 2017 | Further Pure 2 | Matrix Algebra | Cayley-Hamilton Theorem | - |

Edexcel AS/A2 Further Maths 2017 | Further Pure 2 | Matrix Algebra | Cayley-Hamilton Theorem | - |

Methods (UK) | M5 | Matrix algebra | Cayley-Hamilton theorem | - |

OCR A-Level (UK - Pre-2017) | FP1 | Matrix algebra | Cayley-Hamilton theorem | - |

OCR A2 Further Maths 2017 | Pure Core | Further Matrices | Cayley-Hamilton Theorem - Extra | - |

OCR MEI A2 Further Maths 2017 | Extra Pure | Matrices | Cayley-Hamilton Theorem | - |

OCR-MEI A-Level (UK - Pre-2017) | FP2 | Matrix algebra | Cayley-Hamilton theorem | - |

Universal (all site questions) | M | Matrix algebra | Cayley-Hamilton theorem | - |

WJEC A-Level (Wales) | FP1 | Matrix algebra | Cayley-Hamilton theorem | - |