In Linear algebra, the matrix companion of the unit Polynomial

$p\; (T)\; =c\_0\; +\; c\_1\; T\; +\; \backslash \; dowries\; +\; c\_\; \{n-1\}\; t^\; \{n-1\}\; +\; t^n\; \backslash ,$

is in the following way definite square matrix:

$C\; (p)\; =\; \backslash \; begin\; \{bmatrix\}$

0 & 0 & \ dowries & 0 & - c_0 \ \ 1 & 0 & \ dowries & 0 & - c_1 \ \ 0 & 1 & \ dowries & 0 & - c_2 \ \ \ vdots & \ vdots & \ vdots & \ vdots & \ vdots \ \ 0 & 0 & \ dowries & 1 & - c_ {n-1} \ \ \end{bmatrix}.

(even if some consider that it is about transposed of this matrix).

The Polynomial characteristic as well as the minimal Polynôme of C ( p ) is equal to p ; in this direction, the matrix C ( p ) is the “partner” of the polynomial p .

If the polynomial p ( T ) has N distinct roots λ₁,…, λ _N (eigenvalues of C ( p )), then C ( p ) is diagonalisable in the following way:

$V\; C\; (p)\; V^\; \{-\; 1\}\; =\; \backslash \; mbox\; \{diag\}\; (\backslash \; lambda\_1,\; \backslash \; dowries,\; \backslash \; lambda\_n)\; \backslash ,$

where V is the Matrice of Vandermonde associated with λ₁,…, λ _N .

If has is a matrix of order N whose coefficients belong to a body K , then the following proposals are equivalent:

• has is similar to a matrix companion with coefficients in K

• the polynomial characteristic of has is the minimal polynomial of has
• it exists a vector v in K ^N such as { v , has v, has ² v ,…, has ^{N -1} v } is a bases K ^N

All the square matrices are not similar to a matrix companion but any matrix is similar to a matrix made up of blocks of matrices companions. Moreover, these matrices companions can be selected so that their characteristic polynomials divide between them; they are then given in a single way by has . It is the rational canonical form of has .

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