We discuss Jordan bases and the fact that an operator can be put into Jordan canonical form if its characteristic and minimal polynomials factor into linear polynomials. We demonstrate this with an example and provide several exercises.

## Jordan Basis

A basis of $V$ is called a ** Jordan basis** for $T$ if with respect to this basis $T$ has block diagonal matrix $$ \begin{bmatrix} A_1 & & 0 \\ & \ddots & \\ 0 & & A_m \\ \end{bmatrix} $$ where each $A_j$ is an upper triangular matrix of the form $$ A_j = \begin{bmatrix} \lambda_j & 1 & & 0 \\ & \ddots & \ddots & \\ & & \ddots & 1 \\ 0 & & & \lambda_j \end{bmatrix} $$ where the diagonal is filled with some eigenvalue $\lambda_j$ of $T.$

Because there exist operators on real vector spaces that have no eigenvalues, there exist operators on real vector spaces for which there is no corresponding Jordan basis.

Recall that every non-trivial linear operator on a finite-dimensional complex vector space has an eigenvalue.

**Theorem**. Suppose $V$ is a complex vector space. If $T\in \mathcal{L}(V)$, then there is a basis of $V$ that is a Jordan basis for $T.$

An operator $T$ can be put into Jordan canonical form if its characteristic and minimal polynomials factor into linear polynomials. This is always true if the vector space is complex.

## Jordan’s Theorem

**Theorem**. Let $T\in\mathcal{L}(V)$ whose characteristic and minimal polynomials are, respectively, $$ c(t)=(t-\lambda_1)^{n_1} \cdots (t-\lambda_r)^{n_r}) \quad \text{and} \quad m(t)=(t-\lambda_1)^{m_1} \cdots (t-\lambda_r)^{m_r}) $$ where the $\lambda_i$ are distinct scalars. Then $T$ has block diagonal matrix representation $J$ whose diagonal entries are of the form $$ J_{ij}= \begin{bmatrix} \lambda_i & 1 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_i & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_i & 1 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_i \end{bmatrix} $$ For each $\lambda_i$ the corresponding blocks have the following properties:

(1) There is at least one $J_{ij}$ of order $m_i$; all other $J_{ij}$ are of order $\leq m_i$

(2) The sum of the orders of the $J_{ij}$ is $n_i.$

(3) The number of $J_{ij}$ equals the geometric multiplicity of $\lambda_i.$

(4) The number of $J_{ij}$ of each possible order is uniquely determined by $T.$

## Jordan Canonical Form

The matrix $J$ in the above proposition is called the ** Jordan canonical form** of the operator $T.$ A diagonal block $J_{ij}$ is called a

**belonging to the eigenvalue $\lambda_i.$ Obverse that \begin{align*} & \begin{bmatrix} \lambda_i & 1 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_i & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_i & 1 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_i \end{bmatrix} \\ & = \begin{bmatrix} \lambda_i & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_i & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_i & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_i \end{bmatrix} + \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 0 \end{bmatrix} \end{align*} That is, $J_{ij}=\lambda_i I+N$ where $N$ is the nilpotent block.**

*Jordan block*## Example

**Example**. Suppose the characteristic and minimum polynomials of an operator $T$ are, respectively, $$ c(t)=(t-2)^4(t-3)^3 \qquad \text{and} \qquad m(t)=(t-2)^2(t-3)^2. $$ Then the Jordan canonical form of $T$ is one of the following matrices: $$ \begin{bmatrix} 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 \end{bmatrix} \qquad \text{or} \qquad \begin{bmatrix} 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 \end{bmatrix} $$ The first matrix occurs if $T$ has two independent eigenvectors belonging to the eigenvalue 2; and the second matrix occurs if $T$ has three independent eigenvectors belonging to 2.

## Exercises on Canonical Forms (Part A)

**Exercise**. Suppose $N\in \mathcal{L}(V)$ is nilpotent. Prove that the minimal polynomial of $N$ is $z^{m+1}$, where $m$ is the length of the longest consecutive string of $1’\text{s}$ that appears on the line directly above the diagonal in the matrix of $N$ with respect to any Jordan basis for $N.$

**Exercise**. Suppose $V$ is a complex vector space and $T\in \mathcal{L}(V).$ Prove that there does not exist a direct sum decomposition of $V$ into two proper subspaces invariant under $T$ if and only if the minimal polynomial of $T$ is of the form $(z-\lambda)^{\mathop{dim} V}$ for some $\lambda \in \mathbb{C}.$

**Exercise**. Suppose $T\in \mathcal{L}(V)$ and $(v_1,\ldots,v_n)$ is a basis of $V$ that is a Jordan basis for $T.$ Describe the matrix of $T$ with respect to the basis $(v_n,\ldots,v_1)$ obtained by reversing the order of the $v$’s.

**Exercise**. Suppose $A$ is a block diagonal matrix $$ A=\begin{bmatrix} A_1 & & 0 \\

& \ddots & \\ 0 & & A_m \end{bmatrix}, $$ where each $A_j$ is a square matrix. Prove that the set of eigenvalues of $A$ equals the union of the eigenvalues of $A_1,\ldots,A_m.$

**Exercise**. Suppose $A$ is a block upper-triangular matrix $$ A= \begin{bmatrix} A_1 & & * \\ & \ddots & \\ 0 & & A_m \end{bmatrix}, $$ where each $A_j$ is a square matrix. Prove that the set of eigenvalues of $A$ equals the union of the eigenvalues of $A_1$,\ldots,$A_m.$

## Exercises on Canonical Forms (Part B)

**Exercise**. Suppose $V$ is a real vector space and $T\in \mathcal{L}(V).$ Suppose $\alpha, \beta \in \mathbb{R}$ are such that $T^2+\alpha T+\beta I=0.$ Prove that $T$ has an eigenvalue if and only if $\alpha^2 \geq 4 \beta.$

**Exercise**. Suppose $V$ is a real inner-product space and $T\in \mathcal{L}(V).$ Prove that there is an orthonormal basis of $V$ with respect to which $T$ has a block upper-triangular matrix $$ \begin{bmatrix} A_1 & & * \\ & \ddots & \\ 0 & & A_m \end{bmatrix}. $$ where each $A_j$ is a 1-by-1 matrix or a 2-by-2 matrix with no eigenvalues.

**Exercise**. Prove that if $T\in \mathcal{L}(V)$ and $j$ is a positive integer such that $j \leq \mathop{dim} V$, then $T$ has an invariant subspace whose dimension equals $j-1$ or $j.$

## Exercises on Canonical Forms (Part C)

**Exercise**. Prove that there does not exist an operator $T\in \mathcal{L}(\mathbb{R}^7)$ such that $T^2+T+I$ is nilpotent.

**Exercise**. Give an example of an operator $T\in \mathcal{L}(\mathbb{C}^7)$ such that ${T^2+T+I}$ is nilpotent.

**Exercise**. Suppose $V$ is a real vector space and $T\in \mathcal{L}(V).$ Suppose $\alpha, \beta \in \mathbb{R}$ are such that $\alpha^2< 4\beta.$ Prove that null $(T^2+\alpha T + \beta I)^k$ has even dimension for every positive integer $k.$

**Exercise**. Suppose $V$ is a real vector space and $T\in \mathcal{L}(V).$ Suppose $\alpha, \beta \in \mathbb{R}$ are such that $\alpha^2< 4\beta$ and $T^2+\alpha T+\beta I$ is nilpotent. Prove that $\mathop{dim} V$ is even and $(T^2+\alpha T+\beta I)^{\mathop{dim} V/2}=0.$

**Exercise**. Prove that if $T\in \mathcal{L}(\mathbb{R}^3)$ and 5, 7 are eigenvalues of $T$, then $T$ has no eigenpairs.

**Exercise**. Suppose $V$ is a real vector space with $\mathop{dim} V =n $ and $T\in \mathcal{L}(V)$ is such that null $T^{n-2}\neq $ null $T^{n-1}.$ Prove that if $T$ has at most two distinct eigenvalues and that $T$ has no eigenpairs.

**Exercise**. Prove that 1 is an eigenvalue of every square matrix with the property that the sum of the entries in each row equals 1.

**Exercise**. Suppose $V$ is a real vector space with $\mathop{dim} V =2 .$ Prove that if $$ \begin{bmatrix} a & c \\ b & d \end{bmatrix} $$ is the matrix of $T$ with respect to some basis of $V$, then the characteristic polynomial of $T$ equals $(z-a)(z-d)-b c.$

**Exercise**. Suppose $V$ is a real inner-product space and $S\in \mathcal{L}(V)$ is an isometry. Prove that if $(\alpha, \beta)$ is an eigenpair of $S$, then $\beta=1.$