Companion Matrix

In linear algebra, the Frobenius companion matrix of the monic polynomial

p(t)=c_0 + c_1 t + \cdots + c_{n-1}t^{n-1} + t^n ~,

is the square matrix defined as

C(p)=\begin{bmatrix} 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \end{bmatrix}.

With this convention, and writing the basis as, one has (for ), and generates V as a -module: C cycles basis vectors.

Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear recursive relations.

Read more about Companion Matrix:  Characterization, Diagonalizability, Linear Recursive Sequences

Famous quotes containing the words companion and/or matrix:

    “Ambition is an uncomfortable companion many times. He creates a discontent with present surroundings and achievements; he is never satisfied but always pressing forward to better things in the future. Restless, energetic, purposeful, it is ambition that makes of the creature a real man.”
    —Lyndon Baines Johnson (1908–1973)

    “As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
    —Margaret Atwood (b. 1939)