# matrix construction

Given any matrix $A$, can one construct a matrix $B$ such that

1. $B$ is nonnegative and the spectral radius of $B$ is strictly less than 1

2. the determinant of $A$ is equal to the first entry of $B^*$ where $B^*=(I-B)^{-1}$.

Here, by constructing, I certainly do NOT mean "trivial" ones like first computing the determinant of $A$. From the complexity perspective, I want a Logspace algorithm.

Thanks.

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What do you mean by the "first entry" of a matrix? – Gerry Myerson Jul 17 '12 at 0:03
This would yield an O(log(n)) algorithm for finding the determinant of a square matrix. Is this possible? – ncmathsadist Jul 17 '12 at 0:18
@ncmathsadist: How so? You'd still need to invert the matrix to get the determinant. (also I don't think maomao is looking for O(log n) time) Still, I'm inclined to think that such an algorithm does not exist. maomao, do you know any similar examples? What leads you to suspect there is such an algorithm? – us2012 Jul 17 '12 at 0:29
What does he mean by "logspace"? The stipulation $O(\log(n))$ is my best guess. – ncmathsadist Jul 17 '12 at 0:32