Jordan Normal Form, Linear Algebra

Question

Can someone please suggest a step-by-step method for finding the Jordan Normal Form of an nxn matrix.

(An example would be really helpful please)

Thanks!

Answer 1

1) Find the eigenvalues $\{\lambda_j\}_{j=1}^k$, where $1\leq k\leq n$, associated with the matrix $A$.

2) Find the corresponding eigenvectors $\{v_j\}_{j=1}^k$ where $1\leq k\leq n$.

3) If you have $k=n$, diagonalize the matrix by letting $P$ be the matrix whose $j^{th}$ column is $v_j$ for $j=1$ to $n$ and calculating $P^{-1}AP=D=J$ where the Jordan normal form $J$ is diagonal. If not, proceed to step 4.

4) For each eigenvector $v_j$ you found in step 2, construct a Jordan chain of generalized eigenvectors, c.f. https://en.wikipedia.org/wiki/Generalized_eigenvector for several examples.

5) Let $P$ be the matrix whose columns are made up of correctly ordered Jordan chains, then $A=PJP^{-1}$ where $J$ is the almost diagonal Jordan normal form.

Answer 2

Here´s the algorithm I know. But a health warning: until you get to reasonably big matrices there is almost always some easier trick. It calculates the JCF, but does not give an explicit similarity.

So suppose $A$ is an $n\times n$ matrix, with characteristic polynomial $\chi_A(X)$, all of whose roots lie in the field we are working in. (You've got to solve the equation, the algorithm for the JCF only starts once you've done that.)

Now for each of the eigenvalues $\lambda$ we proceed as follows. For $j=1, \dots, n$ we denote by $a_j$ the number of Jordan $j\times j$ blocks associated with $\lambda$: these numbers are what we need to determine. Let $N$ denote for convenience the size of the mega-block corresponding to Jordan blocks of the other eigenvalues.

Calculate (by Gauss Elimination) $r_j:=\text{rank}(A-\lambda I)^{j}$.

Then we have the following equations $$ \begin{align} r_0&=a_1+2 a_2+3 a_3+ \dots +n a_n +N\\ r_1&=a_2+2 a_3+ \dots +(n-1) a_n +N\\ r_2&=a_3+2 a_4+ \dots +(n-2) a_n +N\\ \vdots\\ r_n&=N\\ \end{align} $$ as may be seen at once from the JCF of $A$: similar matrices have the same rank.

These equations determine the number of Jordan blocks of each size and incidentally $N$.

I don't have an example to hand, I'm sorry: and as I say a "typical" example has got to be $10\times 10$ at least!