Demostraciones

Dos matrices semejantes tienen el mismo polinomio mínimo

Demostración

Es consecuencia de que si B= P −1 AP MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2da9iaadcfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWGbbGaamiuaaaa@3C09@   y f(x) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@392E@   es un polinomio, f(B)=f( P −1 AP)= P −1 f(A)P MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGcbGaaiykaiabg2da9iaadAgacaGGOaGaamiuamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaadgeacaWGqbGaaiykaiabg2da9iaadcfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWGMbGaaiikaiaadgeacaGGPaGaamiuaaaa@482A@

Por tanto, el conjunto de polinomios que anula a B es el mismo que el que anula a la matriz A.