Por el resultado anterior si la matriz es diagonalizable su polinomio mínimo es de la forma
(x−
t
1
)⋯(x−
t
i
)⋯(x−
t
r
),
t
i
≠
t
j
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabl+UimjaacIcacaWG4bGaeyOeI0IaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqWIVlctcaGGOaGaamiEaiabgkHiTiaadshadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiilaiaadshadaWgaaWcbaGaamyAaaqabaGccqGHGjsUcaWG0bWaaSbaaSqaaiaadQgaaeqaaaaa@5067@
Derivando
∏
j≠1
(x−
t
j
)
+⋯+
∏
j≠i
(x−
t
j
)
+⋯+
∏
j≠r
(x−
t
j
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebuaeaacaGGOaGaamiEaiabgkHiTiaadshadaWgaaWcbaGaamOAaaqabaGccaGGPaaaleaacaWGQbGaeyiyIKRaaGymaaqab0Gaey4dIunakiabgUcaRiabl+UimjabgUcaRmaarafabaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaadQgaaeqaaOGaaiykaaWcbaGaamOAaiabgcMi5kaadMgaaeqaniabg+GivdGccqGHRaWkcqWIVlctcqGHRaWkdaqeqbqaaiaacIcacaWG4bGaeyOeI0IaamiDamaaBaaaleaacaWGQbaabeaakiaacMcaaSqaaiaadQgacqGHGjsUcaWGYbaabeqdcqGHpis1aaaa@5E80@
que carece de factores comunes con
(x−
t
1
)⋯(x−
t
i
)⋯(x−
t
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabl+UimjaacIcacaWG4bGaeyOeI0IaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqWIVlctcaGGOaGaamiEaiabgkHiTiaadshadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiilaiaadshadaWgaaWcbaGaamyAaaqabaGccqGHGjsUcaWG0bWaaSbaaSqaaiaadQgaaeqaaaaa@5067@
Recíprocamente, por estar en el campo complejo el polinomio mínimo será de la forma
(x−
t
1
)
m
1
⋯
(x−
t
i
)
m
i
⋯
(x−
t
r
)
m
r
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykamaaCaaaleqabaGaamyBamaaBaaameaacaaIXaaabeaaaaGccqWIVlctcaGGOaGaamiEaiabgkHiTiaadshadaWgaaWcbaGaamyAaaqabaGccaGGPaWaaWbaaSqabeaacaWGTbWaaSbaaWqaaiaadMgaaeqaaaaakiabl+UimjaacIcacaWG4bGaeyOeI0IaamiDamaaBaaaleaacaWGYbaabeaakiaacMcadaahaaWcbeqaaiaad2gadaWgaaadbaGaamOCaaqabaaaaaaa@5057@
Su derivada es de la forma
(x−
t
1
)
m
1
−1
g(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykamaaCaaaleqabaGaamyBamaaBaaameaacaaIXaaabeaaliabgkHiTiaaigdaaaGccaWGNbGaaiikaiaadIhacaGGPaaaaa@4220@
que posee el factor
(x−
t
1
)
m
1
−1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykamaaCaaaleqabaGaamyBamaaBaaameaacaaIXaaabeaaliabgkHiTiaaigdaaaGccaWGNbGaaiikaiaadIhacaGGPaaaaa@4220@
en común con el polinomio mínimo. Necesariamente, nuestra hipótesis da m1 = 1. Análogamente, con los demás exponentes.
Por tanto, el polinomio mínimo es de la forma
(x−
t
1
)⋯(x−
t
i
)⋯(x−
t
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabl+UimjaacIcacaWG4bGaeyOeI0IaamiDamaaBaaaleaacaWGPbaabeaakiaacMcacqWIVlctcaGGOaGaamiEaiabgkHiTiaadshadaWgaaWcbaGaamOCaaqabaGccaGGPaaaaa@49BF@
y estamos en las condiciones del resultado anterior. La matriz es diagonalizable.
Notemos que hay métodos efectivos para conocer si un polinomio es primo con su derivada.