15.2.10 Adjoint matrix
The comatrix of a square matrix A of size n is the matrix
B defined by AB=det(A)I. The adjoint
matrix Q(x) of A is the comatrix of xI−A. It is a polynomial of degree
n−1 in x having matrix coefficients and satisfies:
(xI−A)Q(x)=det(xI−A)I=P(x)I,
|
where P(x) is the characteristic polynomial of A.
Since the polynomial P(x)I−P(A) (with matrix coefficients) is
also divisible by x I−A (by algebraic identities), this means
that P(A)=0. We also have Q(x)=I xn−1+⋯+B0
where B0 is the comatrix of A (times −1 if n is odd).
The adjoint_matrix
command finds the characteristic polynomial and adjoint of a given matrix.
-
adjoint_matrix takes
A, a square matrix.
- adjoint_matrix(A) returns the list of the
coefficients of P(x) (the characteristic polynomial of A), and the
list of the matrix coefficients of Q(x) (the adjoint matrix of A).
Examples
Let A=[
]. Input:
adjoint_matrix([[4,1,-2],[1,2,-1],[2,1,0]]) |
|
| ⎡
⎢
⎢
⎣ | ⎡
⎣ | 1,−6,12,−8 | ⎤
⎦ | , | ⎡
⎢
⎢
⎣ | | , | | , | | ⎤
⎥
⎥
⎦ | ⎤
⎥
⎥
⎦ |
| | | | | | | | | | |
|
Hence the characteristic polynomial is:
The determinant of A is equal to −P(0)=8.
The comatrix of A is equal to:
Hence the inverse of A is equal to:
The adjoint matrix of A is:
| ⎡
⎢
⎢
⎣ | x2−2 x+1 | x−2 | −2 x+3 |
x−2 | x2−4 x+4 | −x+2 |
2 x−3 | x−2 | x2−6 x+7 |
| ⎤
⎥
⎥
⎦ |
| .
|
Let A=[
]. Input:
adjoint_matrix([[4,1],[1,2]]) |
|
| ⎡
⎢
⎣ | ⎡
⎣ | 1,−6,7 | ⎤
⎦ | , | ⎡
⎢
⎣ | | , | | ⎤
⎥
⎦ | ⎤
⎥
⎦ |
| | | | | | | | | | |
|
Hence the characteristic polynomial P is:
The determinant of A is equal to P(0)=7.
The comatrix of A is equal to
Hence the inverse of A is equal to:
The adjoint matrix of A is: