Eigenvalues, Eigenvectors, and Tensor Invariants Author: John M. Cimbala, Penn State University Latest revision: 20 September 2007 Let unprimedcoordinates x1, x2, and x3denote the originalaxes. Let primedcoordinates x1′, x2′, and x3′denote the rotatedaxes, in this case rotated to the principal axes. x1x2x321b22b23b2bJJG1x′2x′3x′Note: In this diagram and discussion, superscript 2 refers to eigenvalue #2, and is notan exponent. E.g., b2does notmean “bsquared”. 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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