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find the characteristic and minimal polynomial of the matrix

$$B=\begin{pmatrix} a & 1 & 0 & 0 \\ b & 0 & 1 & 0 \\ c & 0 & 0 & 1 \\ d & 0 & 0 & 0 \end{pmatrix}$$

where $a,b,c,d$ are real numbers

I calculated the characteristic polynomial: $x^4 - ax^3 - bx^2 - cx - d$. I don't know how to go about finding the minimal polynomial

thanks for help

Rodrigo de Azevedo
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  • if two matrices are similar they indeed share the same minimal polynomial. https://en.wikipedia.org/wiki/Matrix_similarity – b00n heT May 25 '16 at 11:31
  • Curious that you found the characteristic polynomial easier than the minimal polynomial, as the latter is quite easy for this particular matrix. Hint: show that $PB=\sum_{i=1}^4 p_{4-i}e_i$ when $P=p_0+p_1X+p_2X^2+p_3X^3$ is any polynomial of degree${}<4$, and that $B^4(e_4)=ae_1+be_2+ce_3+de_4$ (the $e_i$ being the standard basis vectors). – Marc van Leeuwen May 25 '16 at 11:49
  • I don't think it is possible to find the minimal polynomial in general. If we take $a$, $b$, $c$, $d$ such that the polynomial is irreducible( for example using Eisenstein's criterion) the minimal polynomial is the same as the characteristic polynomial. If we choose $a=0$, $b=0$, $c=0$, $d=0$, the characteristic polynomial is $x^4=0$ and the minimal polynomial is $x^3=0$. – S. Venkataraman Sep 01 '17 at 17:37

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