Definition and Examples

For

check that $\hat{T}=PTP^{-1}$.

Example 1.4 shows that the only matrix similar to a zero matrix is itself and that the only matrix similar to the identity is itself.

• Show that the $1\!\times\!1$ matrix whose single entry is $2$ is also similar only to itself.

• Is a matrix of the form $cI$ for some scalar $c$ similar only to itself?

• Is a diagonal matrix similar only to itself?

Consider this transformation of $\mathbb{C}^{3}$

and these bases.

We will compute the parts of the arrow diagram to represent the transformation using two similar matrices.

• Draw the arrow diagram, specialized for this case.

• Compute $T={\rm Rep}_{B,B}(t)$.

• Compute $\hat{T}={\rm Rep}_{D,D}(t)$.

• Compute the matrices for the other two sides of the arrow square.

Consider the transformation $t\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$ described by $x^{2}\mapsto x+1$, $x\mapsto x^{2}-1$, and $1\mapsto 3$.

• Find $T={\rm Rep}_{B,B}(t)$ where $B=\langle x^{2},x,1\rangle$.

• Find $\hat{T}={\rm Rep}_{D,D}(t)$ where $D=\langle 1,1+x,1+x+x^{2}\rangle$.

• Find the matrix $P$ such that $\hat{T}=PTP^{-1}$.

Let $T$ represent $t\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ with respect to $B,B$.

We will convert to the matrix representing $t$ with respect to $D,D$.

• Draw the arrow diagram.

• Give the matrix that represents the left and right sides of that diagram, in the direction that we traverse the diagram to make the conversion.

• Find ${\rm Rep}_{D,D}(t)$.

Exhibit a nontrivial similarity relationship by letting $t\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ act in this way,

picking two bases $B,D$, and representing $t$ with respect to them, $\hat{T}={\rm Rep}_{B,B}(t)$ and $T={\rm Rep}_{D,D}(t)$. Then compute the $P$ and $P^{-1}$ to change bases from $B$ to $D$ and back again.

Show that these matrices are not similar.

Explain Example 1.4 in terms of maps.

Are there two matrices $A$ and $B$ that are similar while $A^{2}$ and $B^{2}$ are not similar?

Prove that if two matrices are similar and one is invertible then so is the other.

Show that similarity is an equivalence relation. (The definition given earlier already reflects this, so instead start here with the definition that $\hat{T}$ is similar to $T$ if $\hat{T}=PTP^{-1}$.)

Consider a matrix representing, with respect to some $B,B$, reflection across the $x$-axis in $\mathbb{R}^{2}$. Consider also a matrix representing, with respect to some $D,D$, reflection across the $y$-axis. Must they be similar?

Prove that matrix similarity preserves rank and determinant. Does the converse hold?

Is there a matrix equivalence class with only one matrix similarity class inside? One with infinitely many similarity classes?

Can two different diagonal matrices be in the same similarity class?

Prove that if two matrices are similar then their $k$-th powers are similar when $k>0$. What if $k\leq 0$?

Let $p(x)$ be the polynomial $c_{n}x^{n}+\cdots+c_{1}x+c_{0}$. Show that if $T$ is similar to $S$ then $p(T)=c_{n}T^{n}+\cdots+c_{1}T+c_{0}I$ is similar to $p(S)=c_{n}S^{n}+\cdots+c_{1}S+c_{0}I$.

List all of the matrix equivalence classes of $1\!\times\!1$ matrices. Also list the similarity classes, and describe which similarity classes are contained inside of each matrix equivalence class.

Does similarity preserve sums?

Show that if $T-\lambda I$ and $N$ are similar matrices then $T$ and $N+\lambda I$ are also similar.