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In this note we show that for a given controllable pair ( A, B ) and any λ＞ 0, a gain matrix K can be chosen so that the transition matrix e(A+BK)t of the system x = (A + BK)x decays at the exponential rate e-λt and the overshoot of the transition matrix can be bounded by MλL for some constants M and L that are independent ofλ. As a consequence, for any h ＞0, a gain matrix K can be chosen so that the magnitude of the transition matrix e(A+BK)t can be reduced by -1/2- (orby anygivenportion) over [0, h ] . An interesting applicafon of the result is in the stabilization of switched linear systems with any given switching rate ( see [ 1 ] ).