Take, for instance, the following matrix: [ 12 5 5 − 12 ] How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation?

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Take, for instance, the following matrix:      [                            12                          5                                      5                          −          12                      ]  How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation?

Jenna Farmer · Accepted Answer

If you know what kind of transformation is (reflection/dilatation/rotation) you can determine one or more eigenvalues and eigenvectors directly.For example   A  v  =  1  ⋅  v indicate the line   v  t is a fixed line and if for w⊥v we have   A  w  =  −  1  ⋅  w then   A is a reflection with respect to the line   v  t.In your example   A  =      [                            12                          5                                      5                          −          12                      ]   we find-       λ    1    =  13        v    1    =  (  5  ,  1  )-       λ    1    =  −  13        v    2    =  (  −  1  ,  5  )is indeed a reflection with dilatation with respect to the line       v    1    t.

Take, for instance, the following matrix: [ <mtable rowspacing="4pt" columnspacing="

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