Jordan Canonical Form

In linear algebra, a Jordan canonical form (JCF) or a Jordan normal form is an upper triangular matrix of a unique format called a Jordan matrix which illustrates a linear operator on a finite-dimensional vector space. Such a matrix contains each non-zero off-diagonal entry equivalent to 1, immediately above the main diagonal, i.e., on the super diagonal, and identical diagonal entries to the left and below. In this article, you will learn how to write the Jordan canonical form with the help of an example.

Learn: Linear algebra

Jordan Canonical Form Definition

Let A be an n by n square matrix and is similar to a block diagonal matrix

Let’s understand the process of writing the Jordan canonical form with the help of examples.

Jordan Canonical Form Example 2×2

Consider a matrix

Solution:

Given matrix is:

Let us find the Jordan canonical form J of A.

For A, the characteristic polynomial is given by:

= -t(-2-t) – (1)(-1)

= 2t + t² + 1

= (t + 1)²

So, λ₁ = -1 is the only eigenvalue and it has algebraic multiplicity m₁ = m_A(λ₁) = 2.

Therefore, the sum of the sizes of the Jordan blocks of J is m₁ = 2.

Now,

A + I =

Here, I is the identity matrix of the same order as A.

This can be written as

Thus, it follows that λ1-eigenspace is EA(-1) = {c(1, -1)t: c ∈ C}, in particular v₁ = 1.

So, we got 1 Jordan block.

From the above, we have m₁ = 2 and v₁ = 1.

Therefore, 1 Jordan block of size 2 with eigenvalue λ₁ = -1

Thus,

Now, we need to find T such that T^-1AT = J; (m ≤ 3)

Consider two vectors v₁ and v₂.

AP = PJ

Write P = (v₁|v₂), with v₁, v₂ ∈ C² and AP = (Av₁|Av₂),

and

PJ = (−v₁|v₁ − v₂)

we have to choose v₁, v₂ such that

1) Av₁ = −v₁

2) Av₂ = v₁ − v₂

3) v₁, v₂ are linearly independent if and only if P is invertible

Thus, we get;

And

Similarly, we can find the Jordan canonical form matrix for 3×3 and 4×4 matrices.

Jordan Canonical Form Problems

1. Find an invertible matrix T such that T⁻¹AT is in Jordan canonical form, where
   \(\begin{array}{l}A=\begin{bmatrix}1 & -2 \\2 & 5 \\\end{bmatrix}\end{array} \)
   .
2. Find a matrix T such that T^-1AT is in Jordan canonical form when
   \(\begin{array}{l}A=\begin{bmatrix}3 & 0 & 0 \\0 & 4 & -1 \\0 & 1 & 2 \\\end{bmatrix}\end{array} \)
   .
3. For
   \(\begin{array}{l}A=\begin{bmatrix}-2 & 2 & 1 \\-7 & 4 & 2 \\5 & 0 & 0 \\\end{bmatrix}\end{array} \)
   , find Jordan canonical form T^-1AT = J.

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