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

Aganippe76

Aganippe76

Answered question

2022-07-06

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

Answer & Explanation

Jenna Farmer

Jenna Farmer

Beginner2022-07-07Added 17 answers

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.

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