Lay Chapter 5,
Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
Mathematica has several commands for working with eigenvalues and eigenvectors.
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Since the Eigenvectors command returns a list eigenvectors, those eigenvectors will appear as rows of a matrix.
It is frequently convenient to reassemble them as columns of a matrix.
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The Eigensystem command returns both eigenvalues and eigenvectors.
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Diagonalization of a matrix, A == .Λ.P
An important result from linear algebra takes the form
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when the matrix A is diagonalizable. Let's work towards that result.
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Now, observe.
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Therefore,
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and
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Another Matrix Diagonalization, A == .Λ.P
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Now, observe.
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Therefore,
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and
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Characteristic Polynomial
Calculate the characteristic polynomial of a certain square matrix.
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Roots of ch[λ] are eigenvalues of a.
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Complex Eigenvalues
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Now, observe.
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Therefore,
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and
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Polar Form of a Matrix
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Find the angle of the rotation.
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... and compute the rotation matrix.
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Find the scaling factor.
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Check the decomposition.
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The Rotation Due to a Complex Eigenvalue
If a real 2x2 matrix has a complex eigenvalue, a particular decomposition will display the rotation "hidden within."
See Lay 5.5, pp. 338--341.
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Construct the matrix c.
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Construct the matrix p.
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Check these results.
We should find that m = p.c.Inverse[p] where c is a polar matrix (rotation with scaling).
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Created by Mathematica (February 22, 2005) | ![]() |