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?

  • $\begingroup$ 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? $\endgroup$ Apr 12 '17 at 18:47
  • $\begingroup$ $-1$ 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. $\endgroup$
    – Peter
    Apr 12 '17 at 18:50
  • $\begingroup$ Both, hopefully, but more so generally $\endgroup$ Apr 12 '17 at 18:58
  • $\begingroup$ We can also determine the minimal polynomial. The geometric multiplicity is $1$, so the minimalpolynomial is equal to the characteristic polynomial, so it is $(x+1)^3$ $\endgroup$
    – Peter
    Apr 12 '17 at 19:03

See Down with Determinants! Each block corresponds to the space of generalized eigenvectors corresponding to some eigenvalue:

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.

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