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I know that the eigenvalue's number is less than the dimension of a matrix, but as the intuition of the eigenvector, each eigenvector keeps the original direction after a linear transform. I think in $\mathbb{R}^n$ there are $n$ vectors which can do this. Why is this not true? Could anyone please give a intuitive explanation of why the eigenvector's number is less than the dimension, or geometric multiplicity is less than algebraic multiplicity?

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A rotation matrix in the plane fixes no vector. So, there is a simple example to supplement your intuition. I'll be interested in what people have to say about the geometry of generalized eigenvectors which I suppose is the true intent of your question. Perhaps begin your post with: "suppose the eigenvalues are all real". –  James S. Cook Dec 3 '12 at 14:30
    
Never heard of the "dimission of a matrix." Googling it turned up nothing. Maybe a translation error? –  Thomas Andrews Dec 3 '12 at 14:35
    
Grammar note: In English, punctuation marks such as , . ? are always followed by a space. I edited to fix it. –  Nate Eldredge Dec 3 '12 at 14:58
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Well, you certainly couldn't have more eigenvalues than independent directions! –  Neal Dec 3 '12 at 15:25
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Less or equal. And, not exactly the 'number' of the eigenvalues, but the sum of the dimensions of the eigenspaces.

The best is to understand it by simple examples.

  1. The identity $\Bbb R^n\to\Bbb R^n$ fixes every vector, so everyone (except $0$) is an eigenvector with eigenvalue $1$ (there are infinitely many of them), spanning the whole space, that is, dimension $n$.
  2. Similarly the reflection about the origo: $x\mapsto -x$ in $\Bbb R^n$: every (nonzero) vector is eigenvector with eigenvalue $-1$.
  3. As James S.Cook commented, the rotation in $\Bbb R^2$ doesn't have any (real) eigenvalue. So this case, it is indeed less.
  4. Take the 'toppling' funtion in the plane: $(x,y)\mapsto (x+y,y)$. Then you can calculate that it has only $1$ as eigenvalue with a $1$ dimensional eigenspace: the $x$-axis.

And, why is the sum of dimensions of eigenspaces of a transformation $A$ is less or equal than the dimension? It is basically because if $\lambda\ne\mu$, then the eigenspaces $E_\lambda:=\{x\mid Ax=\lambda x\}$ and $E_\mu$ are disjoint: $E_\lambda\cap E_\mu=\{0\}$.

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