f*******g
发帖数: 55 | 1 Consider a dynamic process { Q[t] } evolving on the set of M-by-M positive
semi-definite matrices. In particular, { Q[t] } follows the recursive
equation:
Q[t+1] = Q[t] - Q[t] a[t] a[t]' Q[t] / (a[t]' (c I + Q[t] ) a[t]) + d I
where:
a[t] is an M-by-1 vector, which can be adjusted;
c and d are fixed positive real numbers; and
I is an M-by-M identity matrix.
Note that by normalization, we can consider a[t] to be a unit vector without
loss of generality.
The "total reduction" at time t
tr{ Q[t] a[t] a[t]' Q[t] / (a[t]' (c I + Q[t] ) a[t]) }
= a[t]' Q[t] Q[t] a[t] / (a[t]' (c I + Q[t] ) a[t])
is maximized if we choose a[t] to be the eigenvector of Q[t] corresponding
to the largest eigenvalue. Here tr{ A } is the trace of matrix A.
Can we prove that { Q[t] } "decreases" fastest by choosing a[t] in this way
for every t? In particular, can we show that
1. tr{ Q[T] } is minimized for every T > 0
2. 1/T sum_{t=0}^{T-1} tr{ Q[t] } is minimized for every T > 0
3. limsup_{T -> infinity} 1/T sum_{t=0}^{T-1} tr{ Q[t] } is minimized |
|
