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Eigenvalues_and_eigenvectors - Eigenvalues Eigenvectors and...

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Eigenvalues, Eigenvectors, and Tensor Invariants Author: John M. Cimbala, Penn State University Latest revision: 20 September 2007 Let unprimed coordinates x 1 , x 2 , and x 3 denote the original axes. Let primed coordinates x 1 , x 2 , and x 3 denote the rotated axes, in this case rotated to the principal axes. x 1 x 2 x 3 2 1 b 2 2 b 2 3 b 2 b JJG 1 x 2 x 3 x Note : In this diagram and discussion, superscript 2 refers to eigenvalue #2, and is not an exponent. E.g., b 2 does not mean “ b squared”. When the rotated coordinate system aligns with the principal axes , the normal (diagonal) components of the tensor are the eigenvalues , and the off-diagonal components are all zero. The unit vectors pointing in the directions of the principal axes are called the eigenvectors . Each eigenvalue has a corresponding eigenvector. Shown above is the
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