j**********7
发帖数: 4 | 1 An interview question I met.
"Consider a matrix $J$. every eigenvalue of $J$ has positive real part.
Now let us consider infinite product of (1-J/n). Do you think this
infinite product converges? and how about the converge rate?"
Is there anyone can give some advice. thanks! | G******i
发帖数: 163 | 2 Maybe we can think like this:
First, the problem is more or less equivalent to studying the infinite sum
of ln(1-J/n).
But, ln(1-J/n) ~ -J/n for large n. So the real part of this infinite series
should diverge to negative infinity, with the order -c*ln(N).
Taking exponential, we get the infinite product, which should converge to 0,
with the order N^(-c).
Of course, nothing above is written rigorously, but it should not be very
hard to construct a proof along this line. | B********e
发帖数: 10014 | 3 呵呵，特征值对角化更直观
series
0,
【在 G******i 的大作中提到】
: Maybe we can think like this:
: First, the problem is more or less equivalent to studying the infinite sum
: of ln(1-J/n).
: But, ln(1-J/n) ~ -J/n for large n. So the real part of this infinite series
: should diverge to negative infinity, with the order -c*ln(N).
: Taking exponential, we get the infinite product, which should converge to 0,
: with the order N^(-c).
: Of course, nothing above is written rigorously, but it should not be very
: hard to construct a proof along this line.
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