Changing Map Representations

Decide if these are matrix equivalent.

• $\begin{pmatrix}1&3&0\\ 2&3&0\end{pmatrix}$, $\begin{pmatrix}2&2&1\\ 0&5&-1\end{pmatrix}$

• $\begin{pmatrix}0&3\\ 1&1\end{pmatrix}$, $\begin{pmatrix}4&0\\ 0&5\end{pmatrix}$

• $\begin{pmatrix}1&3\\ 2&6\end{pmatrix}$, $\begin{pmatrix}1&3\\ 2&-6\end{pmatrix}$

Which of these are matrix equivalent to each other?

• (a)  $\begin{pmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{pmatrix}$     (b)  $\begin{pmatrix}1&3\\ -1&-3\end{pmatrix}$     (c)  $\begin{pmatrix}-5&1&0\\ -1&0&1\end{pmatrix}$     (d)  $\begin{pmatrix}0&-1\\ 0&5\end{pmatrix}$     (e)  $\begin{pmatrix}1&0&1\\ 2&0&2\\ 1&3&1\end{pmatrix}$     (f)  $\begin{pmatrix}3&1&0\\ 9&3&0\\ -3&-1&0\end{pmatrix}$

Find the canonical representative of the matrix equivalence class of each matrix.

• (a)  $\begin{pmatrix}2&1&0\\ 4&2&0\end{pmatrix}$     (b)  $\begin{pmatrix}0&1&0&2\\ 1&1&0&4\\ 3&3&3&-1\end{pmatrix}$

Suppose that, with respect to

the transformation $t\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is represented by this matrix.

Use change of basis matrices to represent $t$ with respect to each pair.

• $\hat{B}=\langle\begin{pmatrix}0\\ 1\end{pmatrix},\begin{pmatrix}1\\ 1\end{pmatrix}\rangle$, $\hat{D}=\langle\begin{pmatrix}-1\\ 0\end{pmatrix},\begin{pmatrix}2\\ 1\end{pmatrix}\rangle$

• $\hat{B}=\langle\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}1\\ 0\end{pmatrix}\rangle$, $\hat{D}=\langle\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}2\\ 1\end{pmatrix}\rangle$

What sizes are $P$ and $Q$ in the equation $\hat{H}=PHQ$?

Consider the spaces $V=\mathcal{P}_{2}$ and $W=\mathcal{M}_{2\!\times\!2}$, with these bases.

We will find $P$ and $Q$ to convert the representation of a map with respect to $B,D$ to one with respect to $\hat{B},\hat{D}$

• Draw the appropriate arrow diagram.

• Compute $P$ and $Q$.

Find the change of basis matrices $Q$ and $P$ that will convert the representation of a $t\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ with respect to $B,D$ to one with respect to $\hat{B},\hat{D}$.

Find the $P$ and $Q$ to express $H$ via $PHQ$ as a block partial identity matrix.

Use Theorem 2.7 to show that a square matrix is nonsingular if and only if it is equivalent to an identity matrix.

Show that, where $A$ is a nonsingular square matrix, if $P$ and $Q$ are nonsingular square matrices such that $PAQ=I$ then $QP=A^{-1}$.

Why does Theorem 2.7 not show that every matrix is diagonalizable (see Example 2.3)?

Must matrix equivalent matrices have matrix equivalent transposes?

What happens in Theorem 2.7 if $k=0$?

Show that matrix equivalence is an equivalence relation.

Show that a zero matrix is alone in its matrix equivalence class. Are there other matrices like that?

What are the matrix equivalence classes of matrices of transformations on $\mathbb{R}^{1}$? $\mathbb{R}^{3}$?

How many matrix equivalence classes are there?

Are matrix equivalence classes closed under scalar multiplication? Addition?

Let $t\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ represented by $T$ with respect to ${\cal E}_{n},{\cal E}_{n}$.

• Find ${\rm Rep}_{B,B}(t)$ in this specific case.

• Describe ${\rm Rep}_{B,B}(t)$ in the general case where $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$.

• Let $V$ have bases $B_{1}$ and $B_{2}$ and suppose that $W$ has the basis $D$. Where $h\colon V\to W$, find the formula that computes ${\rm Rep}_{B_{2},D}(h)$ from ${\rm Rep}_{B_{1},D}(h)$.

• Repeat the prior question with one basis for $V$ and two bases for $W$.

• If two matrices are matrix equivalent and invertible, must their inverses be matrix equivalent?

• If two matrices have matrix equivalent inverses, must the two be matrix equivalent?

• If two matrices are square and matrix equivalent, must their squares be matrix equivalent?

• If two matrices are square and have matrix equivalent squares, must they be matrix equivalent?

Square matrices are similar if they represent the same transformation, but each with respect to the same ending as starting basis. That is, ${\rm Rep}_{B_{1},B_{1}}(t)$ is similar to ${\rm Rep}_{B_{2},B_{2}}(t)$.

• Give a definition of matrix similarity like that of Definition 2.4.

• Prove that similar matrices are matrix equivalent.

• Show that similarity is an equivalence relation.

• Show that if $T$ is similar to $\hat{T}$ then $T^{2}$ is similar to $\hat{T}^{2}$, the cubes…