Hi,


this is kind of continuation of [this thread](https://mathoverflow.net/questions/6429/asymptotically-multiplicative-functions-and-matrices) to concentrate on a specific problem from linear algebra and analysis that, I think, is rather interesting for itself. 
Here we go:


1) Main problem: Let $(c\_n)\_{n\in\mathbb{N}}$ be some fixed sequence of complex numbers. Is there a sequence of matrices $A\_n=(a(n)\_{ij})\in M\_n(\mathbb{C})$, $n\in\mathbb{N}$, such that $a(n)\_{ij}\widetilde{a(n)\_{ij}}=c\_i c\_j$, where $\widetilde{a(n)\_{ij}}$ denotes the cofactor of $a(n)\_{ij}$? (For a fixed $n\in\mathbb{N}$ this actually establishes a non-linear system of $n^2$ equations with $n^2$ unknowns.)


2) "Inverse" problem: Let $A\_n=(a(n)\_{ij})\in M\_n(\mathbb{C})$, $n\in\mathbb{N}$, be some given sequence of complex quadratic matrices of increasing order. What are necessary and sufficient conditions for such a matrix sequence to have a sequence of complex numbers $(c\_n)\_{n\in\mathbb{N}}$ such that $\forall n\in\mathbb{N}$ $\forall 1\leq i \leq n, 1\leq j \leq n: a(n)\_{ij}\widetilde{a(n)\_{ij}}=c\_i c\_j$ (notation as above)? 


Any input is welcome and I highly appreciate any references to similar or related problems.


Thanks,


efq



factorization of the product of a matrix element and its cofactor
For the main problem, we can use expansion by minors along the ith row to compute


$\det(A\_n) = (-1)^{i+1}(a(n)\_{i1}\widetilde{a(n)\_{i1}} - a(n)\_{i2}\widetilde{a(n)\_{i2}} + a(n)\_{i3}\widetilde{a(n)\_{i3}} - \dots)$


or det(An) = (-1)i+1ci(c1 - c2 + c3 - ...).


This is true for any i <= n, so whenever c1 - c2 + ... - (-1)ncn != 0 we must have ci = (-1)i+1c1 for all i <= n. Therefore when cn != (-1)n+1c1 for the first time, not only must we have c1 - c2 + ... - (-1)ncn = 0 but this alternating sum must vanish for all greater n, meaning that cj = 0 for all j > n.


In conclusion: this is only possible if ck = (-1)k+1c1 for all k < K (K constant but possibly infinite) and then ck = 0 for all k > K.


