Let $A$ by an $n \times n$ matrix over the complex numbers and let $\phi(A,x) = \det(xI-A)$ be the characteristic polynomial of $A$. Let $B_i$ be the principal submatrix of $A$ formed by deleting the $i$-th row and column of $A$. Let $\phi(B_i,x)$ be the characteristic polynomial of $B_i$.
Is it always true that the derivative of $\phi(A,x)$ is the sum of $\phi(B_i,x)$ as $i$ ranges across the rows/columns of $A$?
That is, is the following an identity?
$$ \frac{d\ \phi(A,x)}{dx} = \sum_i \phi(B_i,x) $$
If not, what is a small counterexample? (The equation works for all $2 \times 2$ matrices and all adjacency matrices of graphs….)
Best Answer
It is true.
Proof: $$ \phi(A,x) = \begin{pmatrix} x-a_{1,1} & -a_{1,2} & \cdots & -a_{1,n} \\ -a_{2,1} & x-a_{2,2} & \cdots & -a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{m,1} & -a_{m,2} & \cdots & x-a_{m,n} \end{pmatrix} $$Let us write the derivative using multilinearity: $$ \phi'(A,x) = \begin{pmatrix} 1 &0 & \cdots & 0 \\ -a_{2,1} & x-a_{2,2} & \cdots & -a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{m,1} & -a_{m,2} & \cdots & x-a_{m,n} \end{pmatrix} + \begin{pmatrix} x-a_{1,1} & -a_{1,2} & \cdots & -a_{1,n} \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ -a_{m,1} & -a_{m,2} & \cdots & x-a_{m,n} \end{pmatrix} +\dots + \begin{pmatrix} x-a_{1,1} & -a_{1,2} & \cdots & -a_{1,n} \\ -a_{2,1} & x-a_{2,2} & \cdots & -a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix} = \sum _{i=1}^n \phi(B_i,x) $$