Polynomials of Maps and Matrices

What are the possible minimal polynomials if a matrix has the given characteristic polynomial?

• (a)  $(x-3)^{4}$     (b)  $(x+1)^{3}(x-4)$     (c)  $(x-2)^{2}(x-5)^{2}$     (d)  $(x+3)^{2}(x-1)(x-2)^{2}$

What is the degree of each possibility?

For this matrix

find the characteristic polynomial and the minimal polynomial.

Find the minimal polynomial of this matrix.

Find the minimal polynomial of each matrix.

• (a)  $\begin{pmatrix}3&0&0\\ 1&3&0\\ 0&0&4\end{pmatrix}$     (b)  $\begin{pmatrix}3&0&0\\ 1&3&0\\ 0&0&3\end{pmatrix}$     (c)  $\begin{pmatrix}3&0&0\\ 1&3&0\\ 0&1&3\end{pmatrix}$     (d)  $\begin{pmatrix}2&0&1\\ 0&6&2\\ 0&0&2\end{pmatrix}$     (e)  $\begin{pmatrix}2&2&1\\ 0&6&2\\ 0&0&2\end{pmatrix}$     (f)  $\begin{pmatrix}-1&4&0&0&0\\ 0&3&0&0&0\\ 0&-4&-1&0&0\\ 3&-9&-4&2&-1\\ 1&5&4&1&4\end{pmatrix}$

What is the minimal polynomial of the differentiation operator $d/dx$ on $\mathcal{P}_{n}$?

Find the minimal polynomial of matrices of this form

where the scalar $\lambda$ is fixed (i.e., is not a variable).

What is the minimal polynomial of the transformation of $\mathcal{P}_{n}$ that sends $p(x)$ to $p(x+1)$?

What is the minimal polynomial of the map $\pi\colon\mathbb{C}^{3}\to\mathbb{C}^{3}$ projecting onto the first two coordinates?

Find a $3\!\times\!3$ matrix whose minimal polynomial is $x^{2}$.

What is wrong with this claimed proof of Lemma 1.9: “if $c(x)=\mathchoice{\left|T-xI\right|}{|T-xI|}{|T-xI|}{|T-xI|}$ then $c(T)=\mathchoice{\left|T-TI\right|}{|T-TI|}{|T-TI|}{|T-TI|}=0\,$” ?

Verify Lemma 1.9 for $2\!\times\!2$ matrices by direct calculation.

Prove that the minimal polynomial of an $n\!\times\!n$ matrix has degree at most $n$ (not $n^{2}$ as a person might guess from this subsection’s opening). Verify that this maximum, $n$, can happen.

Show that, on a nontrivial vector space, a linear transformation is nilpotent if and only if its only eigenvalue is zero.

What is the minimal polynomial of a zero map or matrix? Of an identity map or matrix?

What is the minimal polynomial of a diagonal matrix?

A projection is any transformation $t$ such that $t^{2}=t$. (For instance, consider the transformation of the plane $\mathbb{R}^{2}$ projecting each vector onto its first coordinate. If we project twice then we get the same result as if we project just once.) What is the minimal polynomial of a projection?

The first two items of this question are review.

• Prove that the composition of one-to-one maps is one-to-one.

• Prove that if a linear map is not one-to-one then at least one nonzero vector from the domain maps to the zero vector in the codomain.

• Verify the statement, excerpted here, that precedes Theorem 1.8.

  … if a minimal polynomial $m(x)$ for a transformation $t$ factors as $m(x)=(x-\lambda_{1})^{q_{1}}\cdots(x-\lambda_{z})^{q_{z}}$ then $m(t)=(t-\lambda_{1})^{q_{1}}\mathbin{\circ}\cdots\mathbin{\circ}(t-\lambda_{z})^{q_{z}}$ is the zero…

True or false: for a transformation on an $n$ dimensional space, if the minimal polynomial has degree $n$ then the map is diagonalizable.

Let $f(x)$ be a polynomial. Prove that if $A$ and $B$ are similar matrices then $f(A)$ is similar to $f(B)$.

• Now show that similar matrices have the same characteristic polynomial.

• Show that similar matrices have the same minimal polynomial.

• Decide if these are similar.

• Show that a matrix is invertible if and only if the constant term in its minimal polynomial is not $0$.

• Show that if a square matrix $T$ is not invertible then there is a nonzero matrix $S$ such that $ST$ and $TS$ both equal the zero matrix.

• Finish the proof of Lemma 1.7.

• Give an example to show that the result does not hold if $t$ is not linear.

Any transformation or square matrix has a minimal polynomial. Does the converse hold?