Defective Matrices Transforms

Asked Aug 21 '19 at 18:34

Active Aug 21 '19 at 18:34

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Defective Matrix:$\begin{bmatrix}1&1\\0&1\end{bmatrix}$

I am unable to wrap my head around what kind of transforms yield defective matrices or matrices where geometric multiplicity is less than algebric multiplicity. A part that confuses me is the matrix is full rank yet we dont have a basis of eigenvectors possible. I want to know what kind of transforms are these as in projection, reflection or what? What are these transformations that have such a special property(explanations in 2D and 3D).

Here is an example of the type of explanation im looking for but I don't get WHY its true or what causes it:Why are some linear transformations defective? . How do I grok these transformations? It seems a bit un-natural to me.

asked Aug 21 '19 at 18:34

Rahul Deora

You might find this post helpful – Ben Grossmann Aug 21 '19 at 19:56
Another perspective: every matrix can be regarded as a direct sum of Jordan blocks, and every Jordan block can be thought of as a scaled shear map. – Ben Grossmann Aug 21 '19 at 20:02

Defective Matrices Transforms

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