I want to know how to prove that there exists a group of base . these base make the matrix of this map is a matrix whose Principal diagonal element is all zero.
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What is a "group of base"? The trace is invariant under base chance and hence cannot be changed. – Dietrich Burde Oct 27 '19 at 14:07
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@DietrichBurde just a set of base of a linear space – none Yuan Oct 27 '19 at 14:08
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You mean a "basis" of a vector space. If so, the trace cannot be changed. Take $A=\begin{pmatrix} 1 & 0 \cr 0 & 0 \end{pmatrix}$ or $A=(1)$ etc. – Dietrich Burde Oct 27 '19 at 14:09
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@DietrichBurde In other words ，how can prove a matrix whose trace is 0 is similar to a matrix whose all diagonal elements are 0 without Jordan standard form. – none Yuan Oct 27 '19 at 14:15
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You can, before posting this question, just look up here and find a duplicate. – Dietrich Burde Oct 27 '19 at 14:17
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@DietrichBurde thank you very much！ – none Yuan Oct 27 '19 at 14:19
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@DietrichBurde can we prove it from a angle of vector space and linear map？ – none Yuan Oct 27 '19 at 14:23