Bocher’s formula

A colleague of mine mentioned to me today the Bocher’s formula for computing the coefficients of the characteristic polynomial of a matrix. It seems that this formula does not appear too often in textbooks or literature. I’ll just write down the formula and the idea of a simple proof here.

Let the characteristic polynomial of a matrix A be

\displaystyle{p(\lambda)=\lambda^n+a_1\lambda^{n-1}+\cdots+a_n.}

Then the coefficients can be computed by
a_1=-tr(A),
a_2=-\frac{1}{2}\left(a_1tr(A)+tr(A^2)\right),
a_3=-\frac{1}{3}\left(a_2tr(A)+a_1tr(A^2)+tr(A^3)\right),
\vdots
a_n=-\frac{1}{n}\left(a_{n-1}tr(A)+\cdots+a_1tr(A^{n-1})+tr(A^n)\right).

To prove the formula, note that the coefficient a_j is the summation of all possible products of j eigenvalues, i.e.,

\displaystyle{a_j=(-1)^j\sum_{\{t_1\cdots t_j\}\in C_n^j}\lambda_{t_1}\cdots\lambda_{t_j},}

where C_n^j denotes the j-combination of numbers from 1 to n, and the trace of A^i is the sum of the jth power of the eigenvalues, i.e.,

\displaystyle{tr(A^i)=\sum_{t=1}^n\lambda_t^i.}

In addition, we have

\displaystyle{\left(\sum_{\{t_1\cdots t_j\}\in C_n^j}\lambda_{t_1}\cdots\lambda_{t_j}\right)\left(\sum_{t_{j+1}=1}^n\lambda_{t_{j+1}}^i\right)=\sum_{\{t_1\cdots t_{j+1}\}\in C_n^{j+1}}\lambda_{t_1}\cdots\lambda_{t_j}\lambda_{t_{j+1}}^i+\sum_{\{t_1\cdots t_j\}\in C_n^j}\lambda_{t_1}\cdots\lambda_{t_{j-1}}\lambda_{t_j}^{i+1}.}

The above indicates that the first part of (-1)^ja_jtr(A^i) cancels the second part of (-1)^{j-1}a_{j-1}tr(A^{i+1}), whereas the second part of (-1)^ja_jtr(A^i) cancels the first part of (-1)^{j+1}a_{j+1}tr(A^{i-1}). The rest of proof becomes obvious now.

Advertisements

0 Responses to “Bocher’s formula”



  1. Leave a Comment

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s




Categories

Blog Stats

  • 247,883 hits

%d bloggers like this: