Eigenvalues and eigenvectors
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Determining eigenvalues and eigenvectors by use of the characteristic equation
Firstly, we introduce the relevant new terminology. An invariant point is one which is its own image under a transformation represented by a matrix, A. An invariant line is one for which all points on the line map to points also on the line. The points on the line are not necessarily invariant. An eigenvector, x, is a vector that is parallel to an invariant line, so that the invariant line can be described by the equation [Equation goes here - download the original to see it.] Since the line is invariant, but the vector, x, is possibly scaled by the matrix A, then [Equation goes here - download the original to see it.] where l is a scale factor. In fact, this scale factor may be a complex number, but at this level we will examine only matrices that have real valued scale factors. The scale factor is called an eigenvalue. The largest number of eigenvalues and eigenvectors that a matrix can have is equal to its dimension. [Equation goes here - download the original to see it.] It turns out that a matrix can have two distinct eigenvectors but only 1 eigenvalue -that is, the scale factor is repeated or the same for the two eigenvectors. Since not every square matrix describes a scaling, even in the broadest sense given here, it does not follow that we can find eigenvalues and eigenvectors for every square matrix.
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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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