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 be

Then the coefficients can be computed by

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

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

In addition, we have

The above indicates that the first part of cancels the second part of , whereas the second part of cancels the first part of The rest of proof becomes obvious now.

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