## Rank of Jordan canonical form

Hi sir, I was practicing on the 2006 paper and I was wondering what the “rank” meant for Jordan canonical forms. I’d appreciate your help.

Cheers

Xiao

Reply:

Rank is a notion that we use for any linear map, and is defined as the dimension of its image. If the map is given by a matrix, it is also the dimension of the column space. If two matrices represent the same linear map, that is, are similar, then they have the same rank. In particular, the rank of the Jordan canonical form is the same as the rank of the original matrix. One important fact to review is the formula relating the rank and nullity of a linear map.

MK

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