Samuelson–Berkowitz algorithm

The Samuelson-Berkowitz algorithm efficiently computes the characteristic polynomial of an $n\times n$ matrix who entries may be elements of any untial commutative ring without zero divisors.

Description of Algorithm

The Samuelson-Berkowitz algorithm applied to a matrix $A$ produces a vector whose entries are the coefficient of the characteristic polynomial of $A$. It computes this coefficients vector as a recursive product of Toeplitz matrices based on the principal submatrices of $A$

Let $A_{0}$ be an $n\times n$ matrix partitioned so that

$A_{0}=\left[{\begin{array}{c|c}a_{1,1}&R\\\hline C&A_{1}\end{array}}\right]$

The first principal submatrix of $A_{0}$ is the $(n-1)\times (n-1)$ matrix $A_{1}$. Associate with $A_{0}$ the $(n+1)\times n$ Toeplitz matrix $T_{0}$ defined by

$T_{0}=\left[{\begin{array}{c}1\\-a_{1,1}\end{array}}\right]$

if $A_{0}$ is $1\times 1$,

$T_{0}=\left[{\begin{array}{c c}1&0\\-a_{1,1}&1\\-RC&-a_{1,1}\end{array}}\right]$

if $A_{0}$ is $2\times 2$, and in general

$T_{0}=\left[{\begin{array}{c c c c c}1&0&0&0&\cdots \\-a_{1,1}&1&0&0&\cdots \\-RC&-a_{1,1}&1&0&\cdots \\-RA_{1}C&-RC&-a_{1,1}&1&\cdots \\-RA_{1}^{2}C&-RA_{1}C&-RC&-a_{1,1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right]$

That is, all super diagonals of $T_{0}$ consist of zeros, the main diagonal consists of $1$s, the first subdiagonal consists of $-a_{1,1}$ and the $k$th subdiagonal consists of $-RA_{1}^{k-2}C$.

The Toeplitz matrix $T_1$ is the Toeplitz matrix associated with the first principal submatrix $A_{1}$, and so on. The Samuelson-Berkowitz algorithm then states that the vector $v$ defined by

$v=T_{0}T_{1}T_{2}\cdots T_{n-1}$

contains the coefficients of the characteristic polynomial of $A_{0}$.