其中??1??0??0?M???0??0???0?14140141400001000010000141401414?0??0???0??0??0???1??从（4.11）式中易得x(n)?Mx,n?1,2,?n0经过计算，矩阵M的特征值和特征向量为1111?6?(1?5)?4??，?5?(1?5)，?2?1，?3?，?1?1，4242?1??0???1??1??0??0??2???6??????????0??0???1???3?e1???, e2???, e3???, e4????0??0??1??3??0??0???2??6??????????????0???1???1????1???1?4(?3??1?1?(?1?e5??4?1(?1??4?1?1?(?3??4?5)???5)??5)????5)??, ?1?4(?3??1?1?(?1?e6??4?1(?1??4?1?1?(?3??4?5)???5)???5)???5)??M对角化，则有x(n)?PDPx,n?1(0)n?1,2,?(4.12)其中：
??1?0??0P???0??0??0?10?11(?3?402?6110?1?3(?1?41013(?1?40?261111?1(?3?415)(?3?4115)(?1?415)(?1?4115)(?3?4?5)???5)??5)????5)???1?0??0???0nD????0???0??010000?1?????2?00n0000?1??4(1?5)???0n00000?1?4(1??0n?1?0???2?000000???????????n???5)????P?1??1??0??0????0???0??0??2313181?2455?202055?202013231?41?12555?5231314112555?513231?81245?205?20520520?0??1???0??0???0???0??当?1?0??0nn??时D???0?0???0因此，当n??00000??10000?00000??00000?00000??00000??时，（4.12）式中x(n)?1?0??0?P??0??0???000000??10000?00000??1(0)?P?x00000?00000??00000??