Proof of Cayley-Hamilton over every field K
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When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
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That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
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Exercise:
Show that Cayley-Hamilton holds for every field K.
Solution:
Proof. We know that textdetB_sigmain mathcalS_ntextsgnsigma B_sigma_... B_nsigma_n * for every ntimes n matrix B over every field K. Longrightarrow P_AxtextdetA-xI_nin KxA_A_...A_nn But from * we actually see that P_Ax when viewed as a polynomial of xA_A_...Ann is a polynomial with coeffs in mathbbR in fact even in mathbbZ. So P_Axin mathbbRxA_A_...A_nn. Consider now the matrix P_AA. Write it as P_AAP_Ax|_xAleftq_ijA_A_...Annright where q_ijin mathbbRA_A_...A_nn forall A_A_...Annin mathbbR^n^. By Lemma q_ij is the polynomial. And this is indepent of the field we wanted to prove Cayley-Hamilton over.
Show that Cayley-Hamilton holds for every field K.
Solution:
Proof. We know that textdetB_sigmain mathcalS_ntextsgnsigma B_sigma_... B_nsigma_n * for every ntimes n matrix B over every field K. Longrightarrow P_AxtextdetA-xI_nin KxA_A_...A_nn But from * we actually see that P_Ax when viewed as a polynomial of xA_A_...Ann is a polynomial with coeffs in mathbbR in fact even in mathbbZ. So P_Axin mathbbRxA_A_...A_nn. Consider now the matrix P_AA. Write it as P_AAP_Ax|_xAleftq_ijA_A_...Annright where q_ijin mathbbRA_A_...A_nn forall A_A_...Annin mathbbR^n^. By Lemma q_ij is the polynomial. And this is indepent of the field we wanted to prove Cayley-Hamilton over.
Meta Information
Exercise:
Show that Cayley-Hamilton holds for every field K.
Solution:
Proof. We know that textdetB_sigmain mathcalS_ntextsgnsigma B_sigma_... B_nsigma_n * for every ntimes n matrix B over every field K. Longrightarrow P_AxtextdetA-xI_nin KxA_A_...A_nn But from * we actually see that P_Ax when viewed as a polynomial of xA_A_...Ann is a polynomial with coeffs in mathbbR in fact even in mathbbZ. So P_Axin mathbbRxA_A_...A_nn. Consider now the matrix P_AA. Write it as P_AAP_Ax|_xAleftq_ijA_A_...Annright where q_ijin mathbbRA_A_...A_nn forall A_A_...Annin mathbbR^n^. By Lemma q_ij is the polynomial. And this is indepent of the field we wanted to prove Cayley-Hamilton over.
Show that Cayley-Hamilton holds for every field K.
Solution:
Proof. We know that textdetB_sigmain mathcalS_ntextsgnsigma B_sigma_... B_nsigma_n * for every ntimes n matrix B over every field K. Longrightarrow P_AxtextdetA-xI_nin KxA_A_...A_nn But from * we actually see that P_Ax when viewed as a polynomial of xA_A_...Ann is a polynomial with coeffs in mathbbR in fact even in mathbbZ. So P_Axin mathbbRxA_A_...A_nn. Consider now the matrix P_AA. Write it as P_AAP_Ax|_xAleftq_ijA_A_...Annright where q_ijin mathbbRA_A_...A_nn forall A_A_...Annin mathbbR^n^. By Lemma q_ij is the polynomial. And this is indepent of the field we wanted to prove Cayley-Hamilton over.
Contained in these collections
| Title | Creator | Matched on |
|---|---|---|
| Cayley-Hamilton theorem algebraic proof | rk | tags |
| QR-decomposition | rk | tags |
| Scalar product and norm | rk | tags |
| Triagonizablitiy and linear factors | rk | tags |
| Scalar product | rk | tags |

