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alinforest

гæ (СÓÐÃûÆø)

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Show that if p, q are scalar polynomials and
P(¦Ë)+q(¦Ë)=h(¦Ë), p(¦Ë)q(¦Ë)=t(¦Ë),
then, for any square matrix A;
P(A)+()=h(A), p(A)q(A)=t(A).
Verify also that if
p(¦Ë)=q(¦Ë)d(¦Ë)+r(¦Ë),
then, for any square matrix A;
p(A)=q(A)d(A)+r(A)

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alinforest

гæ (СÓÐÃûÆø)

ÒýÓûØÌû:
2Â¥: Originally posted by hank612 at 2013-08-15 22:58:01
ÕâÇ£Éæµ½ÄãÈçºÎ¶Ô´ý¶àÏàʶÁË¡£°ÑLambda µ±×÷´¿´âµÄ·ûºÅ£¬Ö»Âú×ã Lamda^i * Lambda^j = Lambda^(i+j), È»ºóÕųÉÏßÐԿռ䣬 ÄÇôÄã°Ñ¾ØÕóA ´øÈë ³É¸Ã·ûºÅ£¬ ÄÇЩʽ×Ó×ÔÈ»³ÉÁ¢¡£

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3Â¥2013-08-16 09:49:22
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hank612

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alinforest: ½ð±Ò+15, лл~~ 2013-08-16 09:33:44
ÕâÇ£Éæµ½ÄãÈçºÎ¶Ô´ý¶àÏàʶÁË¡£°ÑLambda µ±×÷´¿´âµÄ·ûºÅ£¬Ö»Âú×ã Lamda^i * Lambda^j = Lambda^(i+j), È»ºóÕųÉÏßÐԿռ䣬 ÄÇôÄã°Ñ¾ØÕóA ´øÈë ³É¸Ã·ûºÅ£¬ ÄÇЩʽ×Ó×ÔÈ»³ÉÁ¢¡£

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We_must_know. We_will_know.
2Â¥2013-08-15 22:58:01
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hank612

ÖÁ×ðľ³æ (ÖøÃûдÊÖ)

ÒýÓûØÌû:
3Â¥: Originally posted by alinforest at 2013-08-16 09:49:22
¸Ð¾õ¦ËÓ¦¸ÃÊÇAµÄÌØÕ÷Öµ...

Èç¹ûÊÇÌØÕ÷Öµ£¬¾ÍºÜÏóCayley-Hamiltion ¶¨Àí£ºÈç¹ûf(x)ÊǾØÕóAµÄÌØÕ÷¶àÏîʽ£¬ÄÇôf(A)=0. µ±È»ÕⶨÀíÃ²ËÆºÍÄãÐðÊöµÄÐÔÖÊûɶ¹ØÏµ¡£

Cayley-Hamiltion ¶¨ÀíÊÇÒ»¸ö·Ç³£×ÔÈ»µÄÐÔÖÊ¡£Ê×ÏȶÔÒ»¸öJordan¿é³ÉÁ¢£¬ÒòΪÕâʱµÄf(A)ÊÇÃÝÁãµÄ¡£È»ºó£¬ ¶ÔA²»·ÁÉèΪJordan±ê×¼ÐΣ¬¶ÔÈÎÒâÒ»¸öÏòÁ¿v, Õë¶Ô ker( (A-lambda_i)^{k_i} )×÷Ö±ºÍ·Ö½â£¬Ã¿¸ö·ÖÁ¿f(A)×÷ÓÃÉÏÈ¥¶¼ÊÇÁ㣬Òò´Ëf(A)v=0, ¹Êf(A)=0.
We_must_know. We_will_know.
4Â¥2013-08-16 12:55:54
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alinforest

гæ (СÓÐÃûÆø)

ÒýÓûØÌû:
4Â¥: Originally posted by hank612 at 2013-08-16 12:55:54
Èç¹ûÊÇÌØÕ÷Öµ£¬¾ÍºÜÏóCayley-Hamiltion ¶¨Àí£ºÈç¹ûf(x)ÊǾØÕóAµÄÌØÕ÷¶àÏîʽ£¬ÄÇôf(A)=0. µ±È»ÕⶨÀíÃ²ËÆºÍÄãÐðÊöµÄÐÔÖÊûɶ¹ØÏµ¡£

Cayley-Hamiltion ¶¨ÀíÊÇÒ»¸ö·Ç³£×ÔÈ»µÄÐÔÖÊ¡£Ê×ÏȶÔÒ»¸öJordan¿é³ÉÁ¢£¬ÒòΪÕâ ...

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