El interés de las cadenas de Jordan reside en la posibilidad de construir matrices P regulares que las contengan en sus columnas.

Analicemos previamente un

En general, si una matriz P comienza por la cadena

( T 0 ,  T 1 , ... ,  T k−2 , T k−1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadsfadaWgaaWcbaGaaGimaaqabaGccaGGSaGaaeiiaiaadsfadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaeiiaiaac6cacaGGUaGaaiOlaiaabccacaGGSaGaaeiiaiaadsfadaWgaaWcbaGaam4AaiabgkHiTiaaikdaaeqaaOGaaiilaiaadsfadaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaOGaaiykaaaa@498A@

se tiene

(A−tI) T j−1 = T j ,j=1,...,k−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcacaWGubWaaSbaaSqaaiaadQgacqGHsislcaaIXaaabeaakiabg2da9iaadsfadaWgaaWcbaGaamOAaaqabaGccaGGSaGaamOAaiabg2da9iaaigdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadUgacqGHsislcaaIXaaaaa@4AD8@

(A−tI) T k−1 =(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadgeacqGHsislcaWG0bGaamysaiaacMcacaWGubWaaSbaaSqaaiaadUgacqGHsislcaaIXaaabeaakiabg2da9iaacIcacaaIWaGaaiykaaaa@4180@

Luego

A T j−1 =t T j−1 + T j ,j=1,...,k−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadsfadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGaeyypa0JaamiDaiaadsfadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamivamaaBaaaleaacaWGQbaabeaakiaacYcacaWGQbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam4AaiabgkHiTiaaigdaaaa@4C4C@ A T k−1 =t T k−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadsfadaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaOGaeyypa0JaamiDaiaadsfadaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaaaa@3FF6@

Así, AP=A( T 0 ,  T 1 , ... ,  T k−2 , T k−1 ,...)=( T 0 ,  T 1 , ... ,  T k−2 , T k−1 ,...)B=PB MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6A89@

donde B es de la forma

B=( J L 0 M )             J=( t 0 ⋯ 0 0 1 t ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ t 0 0 0 ⋯ 1 t ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6650@

Más precisamente, J es la matriz de orden k cuyos términos son todos cero salvo los de la diagonal que son iguales a t y los de su paralela inferior que son iguales a 1.

Si P es regular, P −1 AP=( J L 0 M ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaadgeacaWGqbGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaamOsaaqaaiaadYeaaeaacaaIWaaabaGaamytaaaaaiaawIcacaGLPaaaaaa@4007@

En la próxima pantalla se garantizará dicha regularidad.

Algunos autores prefieren escribir la cadena de Jordan al revés. Entonces,

A( T k−1 ,  T k−2 , ... ,  T 1 , T 0 ,...)=( T k−1 ,  T k−2 , ... ,  T 1 , T 0 ,...)( J t L 0 M ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6A74@

  

La matriz J^t recibe un nombre especial

Se dice bloque de Jordan de orden m -asociado al valor propio t- a la matriz cuyos términos son todos 0, salvo los de la diagonal que son iguales a t y los de la paralela superior que son iguales a 1.

Se escribe J_m(t).