# what information is encoded in a Jordan block matrix?

For example, what do the columns and rows mean? My professor mentioned something in a brief comment but I got the idea we don't really have to concern ourselves with this so much, but nevertheless I am interested in knowing.

For instance, it can be shown that:

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

Has Jordan form:

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

How do I read (interpret) this matrix? I know that this is giving us information about the eigenvectors, but what specifically?

• Are you interested in what information is encoded in the Jordan form of an arbitrary matrix or are you interested in the information encoded in one Jordan block? Apr 12 '17 at 18:47
• $-1$ is a triple eigenvalue, but there is only one eigenvector belonging to the eigenvalue $-1$ upto multplication with a scalar. In particular, the matrix is not diagonalizable. Apr 12 '17 at 18:50
• Both, hopefully, but more so generally Apr 12 '17 at 18:58
• We can also determine the minimal polynomial. The geometric multiplicity is $1$, so the minimalpolynomial is equal to the characteristic polynomial, so it is $(x+1)^3$ Apr 12 '17 at 19:03

A vector $v\in V$ is called a generalized eigenvector of $T$ if $$(T − \lambda)^kv = 0$$ for some eigenvalue $\lambda$ of $T$ and some positive integer $k$.
The multiplicity of an eigenvalue $\lambda$ of $T$ is defined to be the dimension of the set of generalized eigenvectors of $T$ corresponding to $\lambda$. We see immediately that the sum of the multiplicities of all eigenvalues of $T$ equals $n$, the dimension of $V$ (from Theorem 3.11(a)). Note that the definition of multiplicity given here has a clear connection with the geometric behavior of $T$, whereas the usual definition (as the multiplicity of a root of the polynomial $\det(zI − T))$ describes an object without obvious meaning.