Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $V$ and $ W$ be finite dimensional vector space over $\mathbb R $ and let $T_1 : V \rightarrow V$ and $T_2 : W \rightarrow W$ be linear transformation whose minimal polynomial are $f_1 (x)= x^3+x^2+x+1$ and$f_2 (x)= x^4 - x^2-2$. let $T : V\oplus W \rightarrow V \oplus W$ be linear transformation s.t. $$T(v,w) =(T_1(v),T_2 (w)) $$ minimal polynomial of T is $f(x)$, then deg $f(x)$ =? and nulity T =?

I can't find such $T_1$, $T_2$and $T$ please guide me..

I don't know where to begin... I am stuck on this problem. Can anyone help me please?

share|cite|improve this question
Hint: least common multiple. – 1015 Jun 21 '13 at 19:01
hint: the matrix of $T$ is block diagonal... – yoyo Jun 21 '13 at 19:02
@ yoyo how can I get the matrix of T? – user45799 Jun 22 '13 at 5:35
@ julien how to use least common multiple. please guide me... – user45799 Jun 22 '13 at 14:24

Just to provide the obvious answer to this old question: the minimal polynomial of a linear operator that stabilises each of a pair of complementary subspaces is the (monic) least common multiple of the minimal polynomials of its restriction to those subspaces; this is immediate from the definition.

One easily computes $\gcd(x^4-x^2-2,x^3+x^2+x+1)=x^2+1$, and using the relation $\gcd(a,b)\operatorname{lcm}(a,b)=ab$, the least common multiple of these polynomials is therefore $(x^4-x^2-2)(x^3+x^2+x+1)/(x^2+1)=x^5+x^4-x^3-x^2-2x-2$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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