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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 offdiagonal 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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This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 CIMBALA

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