Eigenvalues and eigenvectors
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Eigenvalues, eigenvectors, matrices and linear transformations
The reason why eigenvalues and eigenvectors are useful, is because matrices describe linear transformations. So what happens to any basis vectors completely describes the whole transformation. This can be expressed by [Equation goes here - download the original to see it.] When eigenvectors exist the following procedure will find them. Firstly, we form what is called the characteristic equation for the matrix. [Equation goes here - download the original to see it.] For example, for our matrix [Equation goes here - download the original to see it.] The characteristic equation is given by [Equation goes here - download the original to see it.] Next, we solve this equation to find the values of l that satisfy it. For a matrix we can expect at most 2 such values, which we will designate l1 and l2. These will be the eigenvalues of the matrix. To illustrate this process on our matrix, we have already seen [Equation goes here - download the original to see it.] Now solving this [Equation goes here - download the original to see it.] In the next stage we observe that since these are eigenvalues for the matrix, then they must satisfy the equation [Equation goes here - download the original to see it.] so we substitute the values of l1 and l2 to obtain at most two eigenvectors. If the eigenvalues are distinct then it can be shown that there will be two distinct eigenvectors. Continuing with our example [Equation goes here - download the original to see it.] Note that at this stage we have obtained two equations in x and y, but we are expecting only one relationship between them. Both equations must describe the same relationship, and if they do not, then an error has been made earlier on in the calculation. In this case, solving either equation gives the relationship [Equation goes here - download the original to see it.] repeating the process for the second eigenvalue, when [Equation goes here - download the original to see it.] So the matrix, A, is the matrix that stretches the x,y plane by a factor of 4 in the direction of [Equation goes here - download the original to see it.] and by a factor of 2 in the direction of
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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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