Ya se ha comentado que si el índice del valor propio t es k, es posible encontrar un vector T cumpliendo

(A−tI) k T=(0)         (A−tI) k−1 T≠(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcadaahaaWcbeqaaiaadUgaaaGccaWGubGaeyypa0JaaiikaiaaicdacaGGPaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaGGOaGaamyqaiabgkHiTiaadshacaWGjbGaaiykamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaGccaWGubGaeyiyIKRaaiikaiaaicdacaGGPaaaaa@5146@

Construiremos los vectores

T, (A−tI)T,  (A−tI) 2 T, ... ,  (A−tI) k−1 T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaacYcacaqGGaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcacaWGubGaaiilaiaabccacaGGOaGaamyqaiabgkHiTiaadshacaWGjbGaaiykamaaCaaaleqabaGaaGOmaaaakiaadsfacaGGSaGaaeiiaiaac6cacaGGUaGaaiOlaiaabccacaGGSaGaaeiiaiaacIcacaWGbbGaeyOeI0IaamiDaiaadMeacaGGPaWaaWbaaSqabeaacaWGRbGaeyOeI0IaaGymaaaakiaadsfaaaa@5391@

Notemos que el último vector es un vector propio asociado al valor propio t

        

Algunos autores, por motivos que quedarán claros posteriormente, prefieren escribir la familia al revés

(A−tI) k−1 T,  (A−tI) k−2 T, ... , (A−tI)T, T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcadaahaaWcbeqaaiaadUgacqGHsislcaaIXaaaaOGaamivaiaacYcacaqGGaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcadaahaaWcbeqaaiaadUgacqGHsislcaaIYaaaaOGaamivaiaacYcacaqGGaGaaiOlaiaac6cacaGGUaGaaeiiaiaacYcacaqGGaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcacaWGubGaaiilaiaabccacaWGubaaaa@556E@

Ello da lugar al siguiente concepto

Dado un valor propio t de A se dice cadena de vectores propios generalizados de longitud k a una familia de vectores ( T 0 ,  T 1 , ... ,  T k−2 , T k−1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadsfadaWgaaWcbaGaaGimaaqabaGccaGGSaGaaeiiaiaadsfadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaeiiaiaac6cacaGGUaGaaiOlaiaabccacaGGSaGaaeiiaiaadsfadaWgaaWcbaGaam4AaiabgkHiTiaaikdaaeqaaOGaaiilaiaadsfadaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaOGaaiykaaaa@498A@   que cumple Tk-1 es un vector propio asociado a t T i+1 =(A−tI) T i ,i=0,...k−2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccqGH9aqpcaGGOaGaamyqaiabgkHiTiaadshacaWGjbGaaiykaiaadsfadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamyAaiabg2da9iaaicdacaGGSaGaaiOlaiaac6cacaGGUaGaam4AaiabgkHiTiaaikdaaaa@4A1A@

Las cadenas de vectores propios generalizados también se denominan cadenas de Jordan.