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I've been reading wikipedia pag of Jordan canonical form, which induces matrices that does not have eigenbasis, i.e. defective matrices. The physical and geometric meaning of normal matrices are pretty clear. But is there any physical or geometric meaning of defective matrices? I don't have an geometric intuition why can not a matrix, seen as a representation of linear transformation, has an eigenbasis.

Thanks,

Chao

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Good question. Geometrically, you can think of defective matrices as linear mappings applying some kind of shearing, that is, points along a certain dimension are displaced by a distance which depends linearly on the position of the point respect to other dimensions. This dependence on other dimensions is what makes defective matrices non-diagonalizable. Hope this answered your question.

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