I am completely stumped at the proof of Theorem 13 in Chapter 6, Hoffman and Kunze. The theorem goes:

Let $T$ be a linear operator on the finite-dimensional vector space $V$ over field $F$. Suppose that the minimal polynomial for $T$ decomposes over $F$ into a product of linear polynomials. Then there is a diagonalizable operator $D$ on $V$ and a nilpotent operator $N$ on $V$ such that

(i) $T = D + N$

(ii) $DN = ND$

The diagonalizable operator $D$ and the nilpotent operator are uniquely determined by (i) and (ii) and each of them is a polynomial in $T$.

The crux was to prove the uniqueness. In the proof given, there was a sentence: "$D'$ and $N'$ commute with any polynomial in $T$; hence they commute with $D$ and with $N$"

Does this mean $DD' = D'D$ and $NN' = N'N$? How did this come about? I am definitely overlooking something obvious...

  • $\begingroup$ Are you sure it doesn't say $D'$ and $N'$ commute with any polynomial in $T$? $\endgroup$ – Robert Israel Apr 10 '14 at 7:17
  • $\begingroup$ Oh yea it does. Typo! $\endgroup$ – mymindcastadrift Apr 10 '14 at 7:18

First a comment: Uniqueness is fairly routine; the crux is in the existence which requires Chinese Remainder Theorem.

Now to your question: yes, $DD=D'D$ and $NN'=N'N$. The theorem constructs $D$ and $N$ as polynomials $T$. So, if we have possible second candidate, then two polynomials in $T$ which give $D$ and $D'$ will commute.

  • $\begingroup$ Oh okay I see. The book constructed the existence of D and N by looking at factors of minimal polynomials. This implied that both D and N are polynomials in T. So the reasoning was actually just one step from the properties of a polynomial. Thanks! Just curious: How does the Chinese Remainder Theorem get involved? $\endgroup$ – mymindcastadrift Apr 10 '14 at 7:24
  • $\begingroup$ I haven't read that book; but if you are looking for a polynomial satisfying some congruence conditions you are appealing to that theorem in the ring $\mathbf{C}[X]$. $\endgroup$ – P Vanchinathan Apr 10 '14 at 8:55

$N^2=0$ where $N$ is a square matrix order $2$, $N$ will be similar with which two matrices and how?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.