Editing Companion matrix (section)

==From linear ODE to first-order linear ODE system==

Similarly to the above case of linear recursions, consider a homogeneous [[linear ODE]] of order ''n'' for the scalar function <math>y = y(t)</math>:
<math display="block">
y^{(n)} + c_{n-1}y^{(n-1)} + \dots + c_{1}y^{(1)} + c_0 y = 0 .
</math>
This can be equivalently described as a coupled system of homogeneous linear ODE of order 1 for the vector function <math>z(t) = (y(t),y'(t),\ldots,y^{(n-1)}(t))</math>:
<math display="block">z' = C(p)^T z</math>
where <math>C(p)^T</math> is the transpose companion matrix for the characteristic polynomial
<math display="block">p(x)=x^n + c_{n-1}x^{n-1}  + \cdots +  c_1 x + c_0 .</math>
Here the coefficients <math>c_i = c_i(t)</math> may be also functions, not just constants. 

If <math>C(p)^T</math> is diagonalizable, then a diagonalizing change of basis will transform this into a decoupled system equivalent to one scalar homogeneous first-order linear ODE in each coordinate.

An inhomogeneous equation
<math display="block">
y^{(n)} + c_{n-1}y^{(n-1)} + \dots + c_{1}y^{(1)} + c_0 y = f(t)
</math>
is equivalent to the system:
<math display="block">
z' = C(p)^T z + F(t)
</math>
with the inhomogeneity term <math>F(t) = (0,\ldots,0,f(t)) </math>.

Again, a diagonalizing change of basis will transform this into a decoupled system of scalar inhomogeneous first-order linear ODEs.