# Jordan Canonical Form and Minimal Polynomial

I was wondering what the relationship between the minimal polynomial and the Jordan Canonical Form is. Given a matrix, all one needs to do is to compute the characteristic polynomial to determine the Jordan Canonical Form, and using a dot diagram (a la Friedberg), it is unique. Thus, I am wondering what additional information the minimal polynomial gives.

For example, we have the following question.

Find all possible $$7 \times 7$$ Jordan Canonical Forms for a matrix with characteristic polynomial $$\chi(t) = t^2(2-t)^3(3-t)(4-t)$$ whose minimal polynomial is the same as the characteristic polynomial.

Solved:

See Chapter 7.3, Problem 13 from Friedberg.

If $$T$$ is in $$\mathcal L(V)$$ and $$\chi_T(t)$$ splits, let $$λ_1, λ_2, \dots, λ_k$$ be the distinct eigenvalues of $$T.$$ For each integer $$1 \leq i \leq k,$$ let $$p_i$$ be the order of the largest Jordan block corresponding to $$λ_i$$ in a Jordan Canonical Form of $$T$$. Then, the minimal polynomial of $$T$$ is $$(t-λ_1)^{p_1}(t-λ_2)^{p_2} \cdots (t-λ_k)^{p_k}.$$

• I see Jordan Canonical Form is a matrix of blocks of complex number $\lambda$ on the diagonal and 1 'above' it, (mathworld.wolfram.com/JordanCanonicalForm.html), would you tell what it's mainly for? – Charlie Chang Jul 31 at 14:47
• @CharlieChang, the Jordan Canonical Form of a matrix whose characteristic polynomial splits (or a matrix over an algebraically closed field) exists and is unique, and similar matrices have the same Jordan Canonical Form. Further, the Jordan Canonical Form is the sum of a diagonal matrix and a nilpotent matrix, so computations with the Jordan Canonical Form are usually much easier than with the original matrix itself. – Carlo Aug 1 at 22:04