Could someone walk me through why saying for a matrix
if ,
then
is an invalid proof of the Cayley Hamilton Theorem?
This topic has been bugging me for a while now. Since there's no official textbook for the linear algebra class that I had, all I (we) have is a printed latex document compiled from various notes taken by students a few years back. For a proof of this theorem I have a one-liner:
, hence
As far as I can see, there's no equating between a scalar and a matrix and also no insertion of a matrix in the diagonal, since we get
where is a matrix polynomial, as defined here Matrix polynomial - Wikipedia, the free encyclopedia and we use this identity Adjugate matrix - Wikipedia, the free encyclopedia
My question is: could this be valid?