Jordan Normal Form - Motivation

Motivation

An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable. Consider the following matrix:

$A= \left[\!\!\!\begin{array}{*{20}{r}} 5 & 4 & 2 & 1 \\ 0 & 1 & -1 & -1 \\ -1 & -1 & 3 & 0 \\ 1 & 1 & -1 & 2 \end{array}\!\!\right].$

Including multiplicity, the eigenvalues of A are λ = 1, 2, 4, 4. The dimension of the kernel of (A − 4I_n) is 1 (and not 2), so A is not diagonalizable. However, there is an invertible matrix P such that A = PJP−1, where

$J = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 4 \end{bmatrix}.$

The matrix J is almost diagonal. This is the Jordan normal form of A. The section Example below fills in the details of the computation.

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