Finding Jordan normal form over \(\C\)

Section 8.2 Finding Jordan normal form over \(\C\)

In this section we give an algorithm to compute the Jordan normal form of a linear map from a finite-dimensional vector space over \(\C\) to itself. We first define Jordan form.

Definition 8.2.1. (Jordan form of a matrix).

A matrix \(A\in M_n(\C)\) is said to be in the Jordan form if \(A\) is similar to the following matrix.

\begin{equation} J=\begin{pmatrix}J_{\lambda_1}\amp\amp\amp\\\amp J_{\lambda_2}\amp\amp\\\amp\amp\ddots\amp\\\amp\amp\amp J_{\lambda_r}\end{pmatrix}\label{Jordan-form-matrix-eq}\tag{8.10} \end{equation}

where \(\lambda_i\) are eigenvalues of \(A\text{,}\) \(J_{\lambda_i}\) is the Jordan block corresponding to \(\lambda_i\) of size \(k_i\) (see Definition 7.5.7), and \(k_1+k_2+\cdots+k_r=n\text{.}\)

We now state the existence of the Jordan form for any square matrix over \(\C\text{.}\)

Theorem 8.2.2. (Existence of the Jordan normal form).

Vector space form: Let \(V\) be a finite-dimensional vector space over \(\C\) and let \(T\colon V\to V\) be a \(\C\)-linear map. Then there exists a basis \(\mathfrak{B}\) of \(V\) such that the matrix of \(T\) with respect to \(\mathfrak{B}\) has Jordan form (see (8.10)).

Matrix form: Let \(A\in M_n(\C)\text{.}\) There is an invertible matrix \(P\in M_n(\C)\) such that \(P^{-1}AP\) has Jordan form (see (8.10)).

The Jordan form of \(T\) or a matrix is unique except for a permutation of the Jordan blocks occurring in (8.10).

We list some facts that are useful in determining Jordan form of a linear map or a matrix. Let \(V\) be a finite-dimensional vector space over \(\C\) and let \(T\colon V\to V\) be a \(\C\)-linear map. We assume that \(\lambda_i\) are eigenvalues of \(T\text{.}\)

Fact 8.2.3. (Eigenvalues and diagonal of Jordan form).

The characteristic polynomial of \(T\text{,}\) \(\chi_T\) gives us eigenvalues and their multiplicities. The algebraic multiplicity of an eigenvalue \(\lambda\) gives us the number of times \(\lambda\) occur in Jordan form.

Fact 8.2.4. (Number of Jordan blocks).

The dimension of each eigenspace corresponding to \(\lambda_i\text{,}\) i.e., the geometric multiplicity of \(\lambda_i\) gives us the number of Jordan blocks corresponding to \(\lambda_i\) in Jordan form.

Fact 8.2.5. (Size of the largest Jordan block).

The multiplicity of \(\lambda_i\) in the minimal polynomial of \(T\) gives us the size of the largest Jordan block corresponding to \(\lambda_i\) occurring in Jordan form of \(T\text{.}\)

For our purposes the above facts will be enough to get Jordan form of a given linear map or a matrix. In the next section we work out a few examples.