Eigenvalues and eigenvectors
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Diagonalisation
Equations are omitted for technical reasons - download the original pdf
The matrix that scales by a factor of 4 along the x-axis, and by a factor of 2 along the y-axis is [Equation goes here - download the original to see it.] Since there are zero entries for D along the oblique diagonal, its eigenvalues can be immediately read. Matrix D has the same eigenvalues as matrix A of our earlier example, where [Equation goes here - download the original to see it.] In fact, there is a sense in which A and D are the same matrix, since, if the basis vectors of the plane that D transforms were the same as the basis vectors as the plane that A transforms, they would perform the same transformation. That is to same, A can be regarded as having been obtained from D by a change of basis. For A the x-axis of D has become the eigenvector [Equation goes here - download the original to see it.] and the y-axis of D has become the eigenvector [Equation goes here - download the original to see it.] where these eigenvectors are still described in the coordinate system of D. Because of this intimate relationship between the matrix A and the matrix D which has only non-zero entries along its main diagonal, we define them to be similar. [Equation goes here - download the original to see it.] The matrix X that enables the matrix A to be represented by its similar diagonal matrix D is effectively a change of basis. Since it maps the basis vectors of D to the eigenvectors of A, it is simply the matrix [Equation goes here - download the original to see it.] This gives a square matrix, because the two eigenvectors are column matrices. For example, the basis transformation for A above is [Equation goes here - download the original to see it.] The inverse matrix is [Equation goes here - download the original to see it.] Hence the matrix A can be represented by [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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