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.
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

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+x²,y = 3-2x+7x³
Examples: y = 2x, y = 1+x²,
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 | - |