Jordan Canonical Form

Any square matrix over the complex numbers is similar to a block-diagonal matrix in which each block is either a single eigenvalue or a so-called Jordan block with ones on the superdiagonal. This representation, the Jordan canonical form, reveals the algebraic and geometric multiplicities of eigenvalues. For a 2×2 matrix $A$, we have two possibilities: either $A$ is diagonalizable and its Jordan form is a diagonal matrix of eigenvalues, or it is defective and takes the form $\left(\begin{array}{cc}\mathrm{\backslash lambda}& 1\\ 0& \mathrm{\backslash lambda}\end{array}\right)$.

The diagonalizable case occurs when $A$ has two distinct eigenvalues or has a repeated eigenvalue with two linearly independent eigenvectors. The defective case arises when a repeated eigenvalue has only one eigenvector. Determining which case applies involves solving the characteristic polynomial $\mathrm{\backslash det}\left(A,-,\mathrm{\backslash lambda\; I}\right)=0$ and analyzing the resulting eigenvectors.

Computational Steps

Given $A=\left[\begin{array}{cc}a{11}_{}& a{12}_{}\\ a{21}_{}& a{22}_{}\end{array}\right]$, we compute the trace $t=a{11}_{}+a{22}_{}$ and determinant $d=a{11}_{}a{22}_{}-a{12}_{}a{21}_{}$. The characteristic equation is ${\mathrm{\backslash lambda}}^2-t\mathrm{\backslash lambda}+d=0$. Its discriminant $\mathrm{\backslash Delta}=t^2-4d$ decides whether the eigenvalues are equal.

If $\mathrm{\backslash Delta}>0$, the eigenvalues are distinct and $A$ is diagonalizable. If $\mathrm{\backslash Delta}=0$, the eigenvalues coincide. In this scenario we examine whether $A-\mathrm{\backslash lambda\; I}$ is the zero matrix. If it is, then $A$ is already a multiple of the identity and diagonalizable. Otherwise, it has only one eigenvector, leading to a single Jordan block.

Applications

The Jordan form clarifies the structure of linear differential equations, matrix exponentials, and similarity transformations. By knowing the Jordan block sizes, one can compute powers of matrices and exponential maps more easily. In control theory and quantum mechanics, such insights simplify systems of linear operators. Understanding the decomposition also highlights when perturbations might cause dramatic changes, since defective matrices are sensitive to small adjustments.

Learning how to compute the Jordan form fosters deeper intuition about eigenvalues, eigenvectors, and their geometric multiplicities. Experiment with various matrices to see how repeated eigenvalues influence the result. This tool can serve as a stepping stone to more advanced linear algebra topics.