0
$\begingroup$

For given $n\times n$ matrix $A$ and gven $1\leq i_1<...<i_k\leq n$ and $1\leq j_1<...<j_k \leq n$ 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

$\endgroup$
2
$\begingroup$

By the Cauchy-Frenet formula for $1\leq i_1<...<i_k \leq n$ we have

$B_{i_1,...,i_k}^{i_1,...,i_k}=\sum_{1\leq j_1<..<j_k \leq n} (P^T)_{i_1,...,i_k}^{j_1,...,j_k} (AP)_{j_1,...,j_k}^{i_1,...,i_k}= $ $ \sum_{1\leq j_1<..<j_k \leq n}\sum_{1\leq l_1<..<l_k \leq n} P_{j_1,...,j_k}^{i_1,...,i_k} A_{j_1,...,j_k}^{l_1,...,l_k}P_{l_1,...,l_k}^{i_1,...,i_k}. $

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.