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2.46.10  Companion matrix of a polynomial : companion

companion takes as argument an unitary polynomial P and the name of its variable.
companion returns the matrix whose characteristic polynomial is P.
If P(x)=xn+an−1xn−1+...+a−1x+a0, this matrix is equal to the unit matrix of size n−1 bordered with [0,0..,0,−a0] as first row, and with [−a0,−a1,....,−an−1] as last column.
Input :

companion(x^2+5x-7,x)

Output :

[[0,7],[1,-5]]

Input :

companion(x^4+3x^3+2x^2+4x-1,x)

Output :

[[0,0,0,1],[1,0,0,-4],[0,1,0,-2],[0,0,1,-3]]

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