Geometric Multiplicity: eignvectors (1)

I had a question regarding the relationship between multiplicity of eigenvalue and eigenvectors.

I am more interested in eigenvalue's multiplicity than the value itself. Because if eigenvalue has multiplicity, the number of independent eigenvectors ``could'' decrease. My favorite property of eigen-analysis is that is a transformation to simpler basis. Here, simpler means a matrix became a scalar. I even have a problem to understand a 2x2 matrix, but a scalar has no problem, or there is no simpler thing than a scalar. Ax = λ x means the matrix A equals λ, what a great simplification!

My question is

If λ has multiplicity, are there still independent eigenvectors for the eigenvalue?

My intuition said no. I can compute an eigenvector to a corresponding eigenvalue. But, I think I cannot compute the independent eigenvectors for one eigenvalue.

For instance, assume 2x2 matrix that has λ = 1,1, how many eigenvectors? one?

Recently I found this is related with diagonalization using eigenvector. My intuition was wrong.

For one eigenvalue, that has multiplicity, there can be multiple eigenvectors.

I will show the example of this one eigenvalue and multiple eigenventors in next article.

Comments

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

Geometric Multiplicity: eignvectors (2)

If eigenvectors of a matrix A are independent, it is a happy property. Because the matrix A can be diagonalized with a matrix S that column vectors are eigenvectors of A . For example, Why this is a happy property of A? Because I can find A's power easily. A^{10} is not a big deal. Because Λ is a diagonal matrix and power of a diagonal matrix is quite simple. A^{10} = SΛ^{10} S^{-1} Then, why if I want to compute power of A ? That is the same reason to find eigenvectors. Eigenvectors are a basis of a matrix. A matrix can be represented by a single scalar. I repeat this again. This is the happy point, a matrix becomes a scalar. What can be simpler than a scalar value. But, this is only possible when the matrix S's columns are independent. Because S^{-1} must be exist. Now I come back to my first question. Is the λ's multiplicity related with the number of eigenvectors? This time I found this has the name. Geometric multiplicity (GM): the number of in...

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