Coefficients of characteristic polynomial of a matrix

For a given matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed from the columns and rows with indices from $J$.

If the characteristic polynomial of A is $x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, then why $$a_k=(-1)^{n-k}\sum_{|J|=n-k}A[J]$$ that is, why is each coefficient the sum of the appropriately sized principal minors of $A$?

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Your expression is missing some signs. But you might be able to prove this by induction. – Scott Carter Mar 23 '11 at 12:16
Thanks, updated the sign problem. – Ahia Cohen Mar 23 '11 at 12:21
Did you find a solution. I am also looking for an answer to this problem. – Dilawar May 12 '12 at 3:26
Found something useful .. www.mcs.csueastbay.edu/~malek/Class/Characteristic.pdf – Dilawar May 12 '12 at 3:39
– Dilawar May 12 '12 at 3:46

Use the fact that $\begin{vmatrix} a & b+e \\ c & d+f \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} a & e \\ c & f \end{vmatrix}$

We can use this fact to separate out powers of $lambda$. Following is an example for $2 \times 2$ matrix. $$\begin{vmatrix} a-\lambda & b \\ c & d-\lambda \end{vmatrix} = \begin{vmatrix} a & b \\ c & d-\lambda \end{vmatrix} + \begin{vmatrix} -\lambda & b \\ 0 & d-\lambda \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + %% \begin{vmatrix} a & 0 \\ c & -\lambda \end{vmatrix} + %% \begin{vmatrix} -\lambda & b \\ 0 & d \end{vmatrix} + \begin{vmatrix} -\lambda & 0 \\ 0 & -\lambda \end{vmatrix}$$

This decompose $det$ expression into sum of various powers of $\lambda$.

Now try it with a $3 \times 3$ matrix and then generalize it.

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One way to see it: $A:V\to V$ induces the (again linear) maps $\wedge^k A:\wedge^k V\to \wedge^k V$. Your formula (restated in an invariant way, i.e. independently of basis) says that $$\det(x-A)=x^n-x^{n-1} Tr A+ x^{n-2} Tr(\wedge^2 A)-...(*)$$ We can conjugate $A$ so that it becomes upper-triangular with diagonal elements $\lambda_i$ ($\lambda_i$'s are the roots of the char. polynomial). Now for upper triangular matrices the formula $(*)$ says that $$(x-\lambda_1)...(x-\lambda_n)=x^n-x^{n-1}(\sum\lambda_i)+x^{n-2}(\sum\lambda_i\lambda_j)-...$$ which is certainly true, hence $(*)$ is true.

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