TABLE OF CONTENTS Theorem 2.1 (Moment-matching connection of one-side Krylov sub space). For the two Krylov subspaces colspV= Kq( A ?1 E, A ?1 b) and colsp W= K q( A ?T E T , A ?T l), defined in (2.65). For the one-side Krylov subspace method, the reduced system obtained by the projection-based reduction matches q moments of the transfer function of the original system defined in (2.66).
Proof: We first rewrite the reduced system (2.57) as
where E r= W T EV, A r= W T AV, b r= W T b, and 
. Then the reduced system transfer function can be written into
We first compare the moment 0 of the reduced system m r 0 with the original system by using the Krylov subspace colsp V= Kq( A ?1 E, A ?1 b).
The critical step in the above derivation is to realize that A ?1 b belongs to the Krylov subspace Kq( A ?1 E, A ?1 b) and, therefore, we can represent A ?1 b= V r 0 and b= AV r 0.
We can also prove m r 0= m 0 by using Krylov subspace colsp W= Kq( A ?T E T , A ?T l).
In this derivation, we utilize the fact that A ?T l
