AQA FP4 - Eigenvalues and Eigenvectors 2

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AQA FP4 - Eigenvalues and Eigenvectors 2
  1. Mohammed Ladak
  2. Screencast-O-Matic.com
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the eigenvectors and values reveals that there is an enlargement in the direction of a line through the origin. the other eigenvector is the axis the shear is parallel to with an eigenvalue of 1. so guess the resultant is enlargement in the direction corresponding to the eigenvalue that isnt 1.

MrJarastamon
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So the special case of an Eigen value of 1 represents a shear always?
How about if we had a Matrix which have eigen values 1 (eigen vector that is not parallel to the axis) and 2 for example. Would this represent a shear and an enlargement?

MrJarastamon
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What about if I only had a single Eigenvalue of 1? or a single eigenvalue of 2?

mathedup
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Try writing a matrix that represents a shear parallel to the x-axis (to keep it simple) and a matrix that represents a stretch (case 1: parallel to the y-axis. Case 2: Parallel to the x-axis). Combine the transformations of the shear and the stretch in each case. What are the eigenvalues/eigenvectors. Can you explain why? Now try to answer your question..

mathedup