Jordan Canonical Form

Do the check for Example 2.4.

Each matrix is in Jordan form. State its characteristic polynomial and its minimal polynomial.

• (a)  $\begin{pmatrix}3&0\\ 1&3\end{pmatrix}$     (b)  $\begin{pmatrix}-1&0\\ 0&-1\end{pmatrix}$     (c)  $\begin{pmatrix}2&0&0\\ 1&2&0\\ 0&0&-1/2\end{pmatrix}$     (d)  $\begin{pmatrix}3&0&0\\ 1&3&0\\ 0&1&3\\ \end{pmatrix}$     (e)  $\begin{pmatrix}3&0&0&0\\ 1&3&0&0\\ 0&0&3&0\\ 0&0&1&3\end{pmatrix}$     (f)  $\begin{pmatrix}4&0&0&0\\ 1&4&0&0\\ 0&0&-4&0\\ 0&0&1&-4\end{pmatrix}$     (g)  $\begin{pmatrix}5&0&0\\ 0&2&0\\ 0&0&3\end{pmatrix}$     (h)  $\begin{pmatrix}5&0&0&0\\ 0&2&0&0\\ 0&0&2&0\\ 0&0&0&3\end{pmatrix}$     (i)  $\begin{pmatrix}5&0&0&0\\ 0&2&0&0\\ 0&1&2&0\\ 0&0&0&3\end{pmatrix}$

Find the Jordan form from the given data.

• The matrix $T$ is $5\!\times\!5$ with the single eigenvalue $3$. The nullities of the powers are: $T-3I$ has nullity two, $(T-3I)^{2}$ has nullity three, $(T-3I)^{3}$ has nullity four, and $(T-3I)^{4}$ has nullity five.

• The matrix $S$ is $5\!\times\!5$ with two eigenvalues. For the eigenvalue $2$ the nullities are: $S-2I$ has nullity two, and $(S-2I)^{2}$ has nullity four. For the eigenvalue $-1$ the nullities are: $S+1I$ has nullity one.

Find the change of basis matrices for each example.

• (a)  Example 2.12     (b)  Example 2.13     (c)  Example 2.14

Find all possible Jordan forms of a transformation with characteristic polynomial $(x-1)^{2}(x+2)^{2}$.

Find all possible Jordan forms of a transformation with characteristic polynomial $(x-1)^{3}(x+2)$.

Find all possible Jordan forms of a transformation with characteristic polynomial $(x-2)^{3}(x+1)$ and minimal polynomial $(x-2)^{2}(x+1)$.

Find all possible Jordan forms of a transformation with characteristic polynomial $(x-2)^{4}(x+1)$ and minimal polynomial $(x-2)^{2}(x+1)$.

Diagonalize these.

• (a)  $\begin{pmatrix}1&1\\ 0&0\end{pmatrix}$     (b)  $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$

Find the Jordan matrix representing the differentiation operator on $\mathcal{P}_{3}$.

Decide if these two are similar.

Find the Jordan form of this matrix.

Also give a Jordan basis.

How many similarity classes are there for $3\!\times\!3$ matrices whose only eigenvalues are $-3$ and $4$?

Prove that a matrix is diagonalizable if and only if its minimal polynomial has only linear factors.

Give an example of a linear transformation on a vector space that has no non-trivial invariant subspaces.

Show that a subspace is $t-\lambda_{1}$ invariant if and only if it is $t-\lambda_{2}$ invariant.

Prove or disprove: two $n\!\times\!n$ matrices are similar if and only if they have the same characteristic and minimal polynomials.

The trace of a square matrix is the sum of its diagonal entries.

• Find the formula for the characteristic polynomial of a $2\!\times\!2$ matrix.

• Show that trace is invariant under similarity, and so we can sensibly speak of the ‘trace of a map’. (Hint: see the prior item.)

• Is trace invariant under matrix equivalence?

• Show that the trace of a map is the sum of its eigenvalues (counting multiplicities).

• …

To use Definition 2.10 to check whether a subspace is $t$ invariant, we seemingly have to check all of the infinitely many vectors in a (nontrivial) subspace to see if they satisfy the condition. Prove that a subspace is $t$ invariant if and only if its subbasis has the property that for all of its elements, $t(\vec{\beta})$ is in the subspace.

Is $t$ invariance preserved under intersection? Under union? Complementation? Sums of subspaces?

Give a way to order the Jordan blocks if some of the eigenvalues are complex numbers. That is, suggest a reasonable ordering for the complex numbers.

Let $\mathcal{P}_{j}(\mathbb{R})$ be the vector space over the reals of degree $j$ polynomials. Show that if $j\leq k$ then $\mathcal{P}_{j}(\mathbb{R})$ is an invariant subspace of $\mathcal{P}_{k}(\mathbb{R})$ under the differentiation operator. In $\mathcal{P}_{7}(\mathbb{R})$, does any of $\mathcal{P}_{0}(\mathbb{R})$, …, $\mathcal{P}_{6}(\mathbb{R})$ have an invariant complement?

In $\mathcal{P}_{n}(\mathbb{R})$, the vector space (over the reals) of degree $n$ polynomials,

and

are the even and the odd polynomials; $p(x)=x^{2}$ is even while $p(x)=x^{3}$ is odd. Show that they are subspaces. Are they complementary? Are they invariant under the differentiation transformation?

A matrix $S$ is the square root of another $T$ if $S^{2}=T$. Show that any nonsingular matrix has a square root.