# What can I deduce from a matrix by its Jordan Canonical Form?

I solved an interesting, but troubling, problem stated as follows:

If $f(x) = (x-2)^3(x+7)^2$ is the characteristic polynomial and $p(x) = (x-2)^2(x+7)$ is the minimal polynomial for a $5$x$5$ complex matrix $A$, what is the Jordan Form of $A$?

I found the Jordan form to be:

$$\begin{bmatrix} 2 & 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & -7 & 0 \\ 0 & 0 & 0 & 0 & -7 \\ \end{bmatrix}$$

Using a combination of Primary Decomposition Theorem, Cyclic Decomposition Theorem, and Caley Hamilton Theorem.

However, my concern is how I would use such a matrix. I don't know what basis I'm using to get this form, nor do I have handy the form of $A$. I believe I could extract some basic quantities like the rank and nullity of $A$ from here, but finding the image and kernel would be effectively useless without being able to convert the vectors to and from standard coordinates as far as I can tell.

I suppose my questions are:

What kind of information is the Jordan Form providing me with here?

Can I, without first finding $A$, extract the basis that I am using (up to ordering) to represent $A$ in this Jordan form?

You can easily extract the following information about a matrix in JCF:

Its rank, its eigenvalues and their algebraic and geometric multiplicites, its determinant, its trace ...

I think that's a lot of info that you cannot get easily from a matrix not in JCF.

Now, basis with eigenvectors and/or generalized eigenvectors you cannot get it, as far as I can see, only from the JCF. In particular, when the matrix is not diagonalizable, as in your example, you'll have generalized eigenvectors (which are not eigenvectors).

BTW, I think most of the usual representations of JCF use the upper triangular form, which menas that $\;1\;$ you put there in entry 2-1 would rather go in entry 1-2.

• Thank you for the reply! It seems to me that the only properties I'm getting that weren't already given by the characteristic polynomial/minimal polynomial are the rank and nullity. Also, I have noticed that is the typical convention for JCF, but the book I am using (Linear Algebra by Hoffman and Kunze) gives an alternative treatment with the lower triangular form, which arises naturally from how they define Cyclic Decomposition Theorem (and in turn, RCF). – infinitylord Dec 5 '17 at 23:16
• @infinitylord The nullity is the exponent of $x$ in the characteristic polynomial. – amd Dec 6 '17 at 0:08
• @amd: What do you mean by the exponent of $x$? The degree of the characteristic polynomial is the dimension of $V$, so I presume that's not what you mean. Do you mean that if $f(x) = x^k g(x)$ where $g$ is some polynomial with a non-zero constant term, then $k = \text{nullity}(A)$? – infinitylord Dec 6 '17 at 1:04
• Yes, the latter. – amd Dec 6 '17 at 1:20