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this is kind of continuation of this thread 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.



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up vote 3 down vote accepted

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.

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