Samuelson's formula is a recurrence relation for the characteristic polynomial of a square matrix that uses only addition, subtraction, and multiplication. It applies to matrices over any commutative ring with a multiplicative identity (Abdeljaoued 1997, p. 25).
Write an 
matrix in block matrix form
as
![A_(r+1)=[A_r S_r; R_r a],](/images/equations/SamuelsonsFormula/NumberedEquation1.svg) |
(1)
|
where 
, 
is a row vector, and 
is a column vector. Define
 |
(2)
|
where 
is the identity matrix and 
. Expansion by cofactors
gives
 |
(3)
|
where 
denotes the adjugate matrix. Expressing this adjugate matrix in terms of the coefficients
of 
yields Samuelson's formula
 |
(4)
|
Starting with 
, successive leading submatrices
give the characteristic polynomial of
the full matrix without division by integers.
This avoids the divisions by 
that occur in recurrences based on the matrix
trace.
