CompM(P) = Companion matrix of polynomial P

CompM(P) = Companion matrix of polynomial P:

First P is converted to a monic polynomial by dividing by the first non-zero leading coefficient. Then the companion matrix to a monic polynomial

P(x) = x^n + p(n−1)*x^(n−1) + ... + p(1)*x + p(0) = {1; p(n−1); ..., p(1); p(0)}

is the n by n square matrix

| 0 0 ... 0 −p(0) |
| 1 0 ... 0 −p(1) |
| 0 1 ... 0 −p(2) |
| ... ................ ... |
| 0 0 ... 1 −p(n−1) |

with ones on the subdiagonal and the last column given by the coefficients of P(x), {−p(0); −p(1); −p(2); ...; −p(n−1)}. The characteristic polynomial of the companion matrix is the same monic polynomial.

See: Companion Matrix -- From MathWorld

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