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 independent eigenvectors
  • Algebratic multiplicity (AM): the number of multiplicity of eigenvalues

There is no rigid relationship between them. There is only an inequality relationship GM <= AM.

For example, a 4x4 matrix's AM = 3 (The number of different λs  is 2.), GM is not necessary to be 2.

By the way, this S is a special matrix and called Hadamard matrix. I wrote a blog entry how to compute this matrix.  This matrix is so special, it is symmetric, orthogonal, and only contains 1 and -1.

The identity matrix is also an example of such matrix. The eigenvalues of 4x4 identity matrix is λ = 1,1,1,1 and eigenvectors are

I took a day to realize this. But Marc immediately pointed this out.

Though, I still think one λ value corresponds to one eigenvector in general. The number of independent eigenvector is the dimension of null space of A - λ I. The eigenvalue multiplicity is based on this as the form of characteristic function. But, I feel I need to study more to find the deep understanding of this relationship.

Anyway, an interesting thing to me is one eigenvalue can have multiple corresponding eigenvectors.

Gilbert Strang, Introduction to Linear Algebra, 4th Ed.

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