Computes the characteristic polynomial of a finite field element \(a\) or a square matrix \(\mathbf{A}\).

Returns¶

For scalar inputs, the degree-\(m\) characteristic polynomial \(c_a(x)\) of \(a\) over \(\mathrm{GF}(p)\). For square \(n \times n\) matrix inputs, the degree-\(n\) characteristic polynomial \(c_A(x)\) of \(\mathbf{A}\) over \(\mathrm{GF}(p^m)\).

Notes¶

An element \(a\) of \(\mathrm{GF}(p^m)\) has characteristic polynomial \(c_a(x)\) over \(\mathrm{GF}(p)\). The characteristic polynomial when evaluated in \(\mathrm{GF}(p^m)\) annihilates \(a\), that is \(c_a(a) = 0\). In prime fields \(\mathrm{GF}(p)\), the characteristic polynomial of \(a\) is simply \(c_a(x) = x - a\).

An \(n \times n\) matrix \(\mathbf{A}\) has characteristic polynomial \(c_A(x) = \textrm{det}(x\mathbf{I} - \mathbf{A})\) over \(\mathrm{GF}(p^m)\). The constant coefficient of the characteristic polynomial is \(\textrm{det}(-\mathbf{A})\). The \(x^{n-1}\) coefficient of the characteristic polynomial is \(-\textrm{Tr}(\mathbf{A})\). The characteristic polynomial annihilates \(\mathbf{A}\), that is \(c_A(\mathbf{A}) = \mathbf{0}\).

References¶

• https://en.wikipedia.org/wiki/Characteristic_polynomial

Examples¶

The characteristic polynomial of the element \(a\).

In [1]: GF = galois.GF(3**5)

In [2]: a = GF.Random(); a
Out[2]: GF(117, order=3^5)

In [3]: poly = a.characteristic_poly(); poly
Out[3]: Poly(x^5 + 2x^4 + 2x^2 + x + 2, GF(3))

# The characteristic polynomial annihilates a
In [4]: poly(a, field=GF)
Out[4]: GF(0, order=3^5)

The characteristic polynomial of the square matrix \(\mathbf{A}\).

In [5]: GF = galois.GF(3**5)

In [6]: A = GF.Random((3,3)); A
Out[6]: 
GF([[ 62,  35,  73],
    [ 77, 105,  84],
    [188,  44,  64]], order=3^5)

In [7]: poly = A.characteristic_poly(); poly
Out[7]: Poly(x^3 + 222x^2 + 166x + 140, GF(3^5))

# The x^0 coefficient is det(-A)
In [8]: poly.coeffs[-1] == np.linalg.det(-A)
Out[8]: True

# The x^n-1 coefficient is -Tr(A)
In [9]: poly.coeffs[1] == -np.trace(A)
Out[9]: True

# The characteristic polynomial annihilates the matrix A
In [10]: poly(A, elementwise=False)
Out[10]: 
GF([[0, 0, 0],
    [0, 0, 0],
    [0, 0, 0]], order=3^5)