Exercise
https://texercises.com/exercise/proof-of-cayley-hamilton-over-every-field-k/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
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.
Report An Error
You are on texercises.com.
reCaptcha will only work on our main-domain \(\TeX\)ercises.com!
Meta Information
\(\LaTeX\)-Code
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.
Contained in these collections

Similar exercises (48)
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
more (43 more)
Attributes & Decorations
Tags
cayley-hamilton, eth, fs23, lineare algebra, proof
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Decoration
Content image