15.2.8  Characteristic polynomial using Hessenberg algorithm

The pcar_hessenberg command finds the characteristic polynomial of a matrix. It computes the polynomial using the Hessenberg algorithm¹ which is more efficient (O(n³) deterministic) if the coefficients of the matrix are in a finite field or use a finite representation like approximate numeric coefficients. Note however that this algorithm behaves badly if the coefficients are, for example, in ℚ.

• pcar_hessenberg takes one mandatory argument and one optional argument:
  • A, a square matrix.
  • Optionally, x, a variable name.
• pcar_hessenberg(A ⟨,x⟩) returns the characteristic polynomial of A. It is written as the list of its coefficients if no variable name was provided or written as an expression with respect to x if there is a second argument.

Examples

Hence, the characteristic polynomial of [

] in ℤ/37 ℤ is x³−6x²+12x−8.