Let (UV) be its inverse; that is,(UV)(ST)=(USUTVSVT)=(In/20n/20n/2In/2).(11)Then together we carry out the same similarity transformation on P1 and P2 as follows:(UV)P1(ST)=(UP1SUP1TVP1SVP1T)=(0n/2UP1T0n/2VP1T),(UV)P2(ST)=(UP2SUP2TVP2SVP2T)=(UP2S0n/2VP2SIn/2),(12)where P1 and P2 are idempotent implies that VP1T and UP2S are idempotent and r(P1) = r(P2) = n/2 implies that VP1T = In/2 and UP2S = 0n/2. Hence, X is similar to the following matrix:(0n/2��UP1T0n/20n/2)+(0n/20n/2?��VP2S0n/2).(13)That is, X is s2N in Mn(K).When car(K) = 2, X is (��, ��) composite for arbitrary nonzero �� K, we can similarly prove that X is s2N in Mn(K) replacing ?�� with �� in the previous proof.Step 4 �� Suppose car(K) �� 2 and all eigenvalues of X are in K; then by Corollary 5, jk(X, ��) = jk(X, ?��) for every k Z+ and arbitrary nonzero �� K.
Moreover, X is similar to X1 Xs, where both the characteristic polynomial and the minimal polynomial of Xi are [(x ? ��i)(x + ��i)]ri = (x2 ? ��i2)ri with 2��i=1sri = n and ��i K0 is one of eigenvalues of X for every i [s]. Without loss of generality, we just need to prove Xi is s2N.Since Xi is similar to C((x2 ? ��i2)ri) as follows:(00??0a010??00010?0a2?????????10a2ri?20??010),(14)where (x2 ? ��i2)ri = x2ri ? a2ri?2x2ri?2 ? ?a2x2 ? a0. We have C((x2 ? ��i2)ri) = E2,1 + +E2ri,2ri?1 + a0E1,2ri + a2E3,2ri + +a2ri?2E2ri?1,2ri = (E2,1 + E4,3 + +E2ri,2ri?1)+(E3,2 + +E2ri?1,2ri?2 + a0E1,2ri + a2E3,2ri + +a2ri?2E2ri?1,2ri) = N1 + N2. Obviously, both N1 and N2 are square nilpotent matrices; that is, Xi is s2N. Hence, X is s2N in Mn(K).
When car(K) = 2, all blocks in the Jordan reduction of X with respect to �� K0 have an even size by Corollary 6; that is, both the characteristic polynomial and minimal polynomial of every block with respect to �� are (x + ��)si = ((x + ��)2)si/2 = (x2 + ��2)si/2, where si is even. Similarly, we can also prove that X is s2N in Mn(K).
The triple-band bandstop filter (TBBSF) has evolved rapidly and has led to a dramatic demand for a lower-cost product with a compact size and strong communication capabilities. In a microwave communication system, the bandstop filter (BSF) is an important component that is typically adopted in the transmitter and receiver system. Triple-band antennas, baluns, and filters are required to accommodate triple-band wireless systems [1�C3].
Triple-band BSFs are the key components for suppressing the specific bands of frequencies among these devices. Compared to single-band and dual-band filters, these filters are more popular due to their miniature size.Recently, an increasing number of researchers have paid attention to triple-band bandstop filters as a key component for evolving WiMAX applications Entinostat of the future. The conventional topology of filters design fails to meet the requirement of compact size [2�C5]. Thus, this paper focuses on a design that expands the frequency band to reduce the size with a rectangular meandered-line stepped impedance technology.