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I want to know if there is any difference between row eigenvectors and column eigenvectors?

By mentioning "difference", I don't mean the exact value of the vectors. I mean the maximal number of linearly independent eigenvectors of the matrix. Or if I will get exactly the same space spanned by both row eigenvectors and column eigenvectors?

Thank you and I appreciate your replies.

janmarqz
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princeward
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1 Answers1

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In general the spaces will be different, but will have the same dimension.

The reason the spaces have the same dimension is because a matrix and its transpose have the same Jordan form. [See for example THIS.]

To get that the spaces are not the same...even after converting rows into columns, write down just about any example. [ Like THIS and THIS.]

Community
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Bill Cook
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  • Perhaps any nonsymmetric example... – zcn Jan 28 '14 at 03:13
  • Thank you so much Bill. I now understand why the dimension will be the same. However, I didn't get why the spaces are not the same. What did you mean in the example? As far as I'm concerned, if the dimension of the eigenvector is same, then the space spanned by them will be the same, although the bases in each space are different. Am I right? – princeward Jan 28 '14 at 03:22
  • What I meant is that the spaces aren't merely "transposes" of each other (turning rows into columns). If by "same" you mean "isomorphic", then yes of course they're the same - any two vector spaces of the same dimension are isomorphic. – Bill Cook Jan 28 '14 at 05:18