Accompanying matrix

The accompanying matrix is a special matrix that can be assigned to a normalized polynomial . Thus a companion matrix is an object from linear algebra .

definition

The companion matrix of a normalized polynomial th degree on a body is the square - Matrix ${\ displaystyle n}$ ${\ displaystyle f (x) = x ^ {n} + a_ {n-1} x ^ {n-1} + \ dots + a_ {1} x + a_ {0}}$ ${\ displaystyle n \ times n}$

{\ displaystyle A (f) = {\ begin {pmatrix} 0 & 0 & \ dots & 0 & -a_ {0} \\ 1 & 0 & \ dots & 0 & -a_ {1} \\ 0 & 1 & \ ddots & \ vdots & -a_ {2} \\ \ vdots & \ ddots & \ ddots & 0 & \ vdots \\ 0 & \ dots & 0 & 1 & -a_ {n-1} \\\ end {pmatrix}}.}

Sometimes the transposed matrix of is also used, but this does not change anything fundamentally. This special form of the matrix is also called cardinal form. ${\ displaystyle A (f)}$

properties

The characteristic polynomial and the minimal polynomial of is even . On the other hand, a matrix is similar to the companion matrix of the characteristic polynomial of if and only if the minimal polynomial and the characteristic polynomial of are identical. ${\ displaystyle A (f)}$ ${\ displaystyle f}$ ${\ displaystyle n \ times n}$ ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle A}$

If the polynomial has exactly different zeros , then it is diagonalizable : for the Vandermonde matrix . ${\ displaystyle f}$ ${\ displaystyle n}$ ${\ displaystyle \ lambda _ {1}, \ dots, \ lambda _ {n}}$ ${\ displaystyle A (f)}$ ${\ displaystyle VA (f) V ^ {- 1} = \ mathrm {diag} (\ lambda _ {1}, \ dots, \ lambda _ {n})}$ ${\ displaystyle V = V (\ lambda _ {1}, \ dots, \ lambda _ {n})}$

The reverse is true of this, that is, an accompanying matrix can be diagonalized if and only if it has exactly different zeros . ${\ displaystyle A (f)}$ ${\ displaystyle f}$ ${\ displaystyle \ mathrm {grad} (f)}$

application

Accompanying matrices appear in normal form theory . The existence of the Frobenius normal form says that every matrix is similar to a block diagonal matrix whose blocks are companion matrices.

Individual evidence

↑ Hans-Joachim Kowalsky, Gerhard O. Michler: Lineare Algebra . de Gruyter, Berlin 2003, ISBN 3-11-017963-6 , p. 349 .
^ Roger A. Horn, Charles R. Johnson: Matrix Analysis. Cambridge University Press, 1990, ISBN 978-0-521-38632-6 , p. 147 ( limited preview in Google book search).

literature

Siegfried Bosch : Linear Algebra. 5th edition, Springer, Berlin / Heidelberg 2014, ISBN 978-3-642-55259-5 , chapter 6.5.

[1] Hans-Joachim Kowalsky, Gerhard O. Michler: Lineare Algebra . de Gruyter, Berlin 2003, ISBN 3-11-017963-6 , p. 349 .

[2] Roger A. Horn, Charles R. Johnson: Matrix Analysis. Cambridge University Press, 1990, ISBN 978-0-521-38632-6 , p. 147 ( limited preview in Google book search).