Jordan Canonical Form

Jordan Canonical Form (JCF) is a generalization of diagonalization. Every square matrix $$A$$ has a JCF, even when it's not diagonalizable. The JCF of a matrix $$A$$ is obtained with a similarity transformation, that is,

$$T^{-1}A T = J = \text{diag}(J_1, \dots, J_q),$$

where $$J$$ is a block-diagonal matrix and each block $$J_i$$ is a bidiagonal matrix in the form:

$$J_i = \begin{bmatrix}\ddots\end{bmatrix}$$