What is a relation between principal minors of $A$ and $P^TAP$?

For given $$n\times n$$ matrix $$A$$ and gven $$1\leq i_1<... and $$1\leq j_1<... we denote by $$A_{i_1,..,i_k}^{j_1,...,j_k}$$ the minor of $$A$$ arisen by deletting in $$A$$ all rows except $$i_1,...,i_k$$ and all columns except $$j_1,...,j_k$$.
Let $$A$$ and $$P$$ be square $$n\times n$$ matices with entries from a some field, where $$A$$- symmetric, $$P$$-nonsingular. Let $$B=P^TAP$$.

What is connection between principal minors of $$B_{i_1,...,i_k}^{i_1,...,i_k}$$ of $$B$$ and minors of $$A$$?

Thanks

By the Cauchy-Frenet formula for $$1\leq i_1<... we have
$$B_{i_1,...,i_k}^{i_1,...,i_k}=\sum_{1\leq j_1<.. $$\sum_{1\leq j_1<..