Let $M_n( \Bbb{C} )$ be set of all $n \times n$ matrices over $$\{\Bbb{C}\setminus \Bbb{R} \}$$ then for $A,B \in M_n( \Bbb{C} )$ are similar iff minimal polynomial and characteristic polynomial of $A,B$ are same? Is this statement true! It is obvious that converse is true but I am not sure about the other way around! I know that above statement is not true if matrix is over $\Bbb{R}$ can we generalise it over $\Bbb{C}$ ? I am asking specifically for matrix with pure complex entries!
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1No, it is not equivalent. Did you search this site already? – Dietrich Burde Dec 08 '19 at 14:09
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The matrix counter-examples are even over $\Bbb Q$, hence also over $\Bbb C$. – Dietrich Burde Dec 08 '19 at 14:22
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Edited my question, please read it again! – Alfha Dec 08 '19 at 14:27
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@DietrichBurde Seems that even $0$ is not allowed… – xbh Dec 08 '19 at 14:33
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You have edited the question too often now. How can someone give then a valid answer? – Dietrich Burde Dec 08 '19 at 14:35
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It is final version! Sorry for trouble! – Alfha Dec 08 '19 at 14:36
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Simple examples. $$ \newcommand{\abs}[1]{\left\vert #1 \right\vert} \newcommand\rme{\mathrm e} \newcommand\imu{\mathrm i} \newcommand\diff{\,\mathrm d} \DeclareMathOperator\sgn{sgn} \renewcommand \epsilon \varepsilon \newcommand\trans{^{\mathsf T}} \newcommand\F {\mathbb F} \newcommand\Z{\mathbb Z} \newcommand\R{\Bbb R} \newcommand \N {\Bbb N} \newcommand\bm\boldsymbol \bm A = \imu \begin{bmatrix} 1 & 1 & & \\ & 1 && \\ &&1&\\ &&&1 \end{bmatrix}, \bm B = \imu\begin{bmatrix} 1 & 1 & & \\ & 1 && \\ &&1&1\\ &&&1 \end{bmatrix}, $$ then both the minimal polynomials are $(x-\imu)^2$ and the characteristic polynolmials are $(x-\imu)^4$, but $\bm A, \bm B$ are not similar.
The statement is true if $n \leqslant 3$.
xbh
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