Eigenvalues and eigenvectors
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Power matrices
An immediate benefit of the technique of diagonalising a given matrix, can be seen in the ability to use diagonal matrices to find higher powers of that matrix. Firstly, let us suppose that we have a diagonal representation of a matrix, [Equation goes here - download the original to see it.] Then [Equation goes here - download the original to see it.] Generalising this result [Equation goes here - download the original to see it.] The proof would be by mathematical induction. Example (i) Find the eigenvalues and eigenvectors of the matrix [Equation goes here - download the original to see it.] and hence find a diagonal matrix such that [Equation goes here - download the original to see it.] (ii) Solution [Equation goes here - download the original to see it.]
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Contents of
Eigenvalues and eigenvectors
1 Eigenvalues and Eigenvectors
2 Determining eigenvalues and eigenvectors by use of the characteristic equation
3 Eigenvalues, eigenvectors, matrices and linear transformations
4 Diagonalisation
5 Power matrices
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