Could someone walk me through why saying for a matrix

if ,

then

is an invalid proof of the Cayley Hamilton Theorem?

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- February 28th 2010, 07:27 PMchiph588@Cayley Hamilton Theorem
Could someone walk me through why saying for a matrix

if ,

then

is an invalid proof of the Cayley Hamilton Theorem? - February 28th 2010, 07:32 PMBruno J.
Well if you look at the matrix you see that occurs within the matrix; with this in mind, it certainly makes no sense to substitute a matrix for (you'd get matrices within a matrix, and only on the diagonal at that!).

- February 28th 2010, 07:33 PMDrexel28
- February 28th 2010, 07:35 PMchiph588@
- February 28th 2010, 07:38 PMtonio
- September 6th 2011, 09:32 AMlesnikRe: 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?