Diagonalization, similarity, and powers of a matrix

1. Find a diagonalization of $A$, if one exists, when .

2. Can the diagonal matrix be diagonalized? If so, explain how to find the matrices $P$ and $D$.

3. Find a diagonalization of $A$, if one exists, when .

4. Find a diagonalization of $A$, if one exists, when .

5. Suppose that $A=PDP^{-1}$ where .

   1. Explain why $A$ is invertible.

   2. Find a diagonalization of $A^{-1}$.

   3. Find a diagonalization of $A^3$.

1. Let's begin with the diagonal matrix . Find the powers $D^2$, $D^3$, and $D^4$. What is $D^k$ for a general value of $k$?

2. Suppose that $A$ is a matrix with eigenvector $\vvec$ and associated eigenvalue $\lambda$; that is, $A\vvec = \lambda\vvec$. By considering $A^2\vvec$, explain why $\vvec$ is also an eigenvector of $A^2$ with eigenvalue $\lambda^2$.

3. Suppose that $A= PDP^{-1}$ where . Remembering that the columns of $P$ are eigenvectors of $A$, explain why $A^2$ is diagonalizable and find a diagonali…

Determine whether the following matrices are diagonalizable. If so, find matrices $D$ and $P$ such that $A=PDP^{-1}$.

1. $A = \left[\begin{array}{rr} -2 & -2 \\ -2 & 1 \\ \end{array}\right]$.

2. $A = \left[\begin{array}{rr} -1 & 1 \\ -1 & -3 \\ \end{array}\right]$.

3. $A = \left[\begin{array}{rr} 3 & -4 \\ 2 & -1 \\ \end{array}\right]$.

4. $A = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 2 & -2 & 0 \\ 0 & 1 & 4 \\ \end{array}\right]$.

5. $A = \left[\begin{array}{rrr} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \\ \end{array}\right]$.

Determine whether the following matrices have complex eigenvalues. If so, find the matrix $C$ such that $A = PCP^{-1}$.

1. $A = \left[\begin{array}{rr} -2 & -2 \\ -2 & 1 \\ \end{array}\right]$.

2. $A = \left[\begin{array}{rr} -1 & 1 \\ -1 & -3 \\ \end{array}\right]$.

3. $A = \left[\begin{array}{rr} 3 & -4 \\ 2 & -1 \\ \end{array}\right]$.

Determine whether the following statements are true or false and provide a justification for your response.

1. If $A$ is invertible, then $A$ is diagonalizable.

2. If $A$ and $B$ are similar and $A$ is invertible, then $B$ is also invertible.

3. If $A$ is a diagonalizable $n\times n$ matrix, then there is a basis of $\real^n$ consisting of eigenvectors of $A$.

4. If $A$ is diagonalizable, then $A^{10}$ is also diagonalizable.

5. If $A$ is diagonalizable, then $A$…

Provide a justification for your response to the following questions.

1. If $A$ is a $3\times3$ matrix having eigenvalues $\lambda = 2, 3, -4$, can you guarantee that $A$ is diagonalizable?

2. If $A$ is a $2\times 2$ matrix with a complex eigenvalue, can you guarantee that $A$ is diagonalizable?

3. If $A$ is similar to the matrix $B = \left[\begin{array}{rrr} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 3 \\ \end{array}\right]$, is $A$ diagonalizable?

4. What can you say about a matrix that is similar to the identity matrix?

5. If $A$ is a diagonalizable $2\times2$ matrix…

Describe geometric effect that the following matrices have on $\real^2$:

1. $A = \left[\begin{array}{rr} 2 & 0 \\ 0 & 2 \\ \end{array}\right]$

2. $A = \left[\begin{array}{rr} 4 & 2 \\ 0 & 4 \\ \end{array}\right]$

3. $A = \left[\begin{array}{rr} 3 & -6 \\ 6 & 3 \\ \end{array}\right]$

4. $A = \left[\begin{array}{rr} 4 & 0 \\ 0 & -2 \\ \end{array}\right]$

5. $A = \left[\begin{array}{rr} 1 & 3 \\ 3 & 1 \\ \end{array}\right]$

We say that $A$ is similar to $B$ if there is a matrix $P$ such that $A = PBP^{-1}$.

1. If $A$ is similar to $B$, explain why $B$ is similar to $A$.

2. If $A$ is similar to $B$ and $B$ is similar to $C$, explain why $A$ is similar to $C$.

3. If $A$ is similar to $B$ and $B$ is diagonalizable, explain why $A$ is diagonalizable.

4. If $A$ and $B$ are similar, explain why $A$ and $B$ have the same characteristic polynomial; that is, explain why $\det(A-\lambda I) = \det(B-\lambda I)$.

5. If $A$ and $B$ are similar,…

Suppose that $A = PDP^{-1}$ where .

1. Explain the geometric effect that $D$ has on vectors in $\real^2$.

2. Explain the geometric effect that $A$ has on vectors in $\real^2$.

3. What can you say about $A^2$ and other powers of $A$?

4. Is $A$ invertible?

When $A$ is a $2\times2$ matrix with a complex eigenvalue $\lambda = a+bi$, we have said that there is a matrix $P$ such that $A=PCP^{-1}$ where $C=\left[\begin{array}{rr} a & -b \\ b & a \\ \end{array}\right]$. In this exercise, we will learn how to find the matrix $P$. As an example, we will consider the matrix $A = \left[\begin{array}{rr} 2 & 2 \\ -1 & 4 \\ \end{array}\right]$.

1. Show that the eigenvalues of $A$ are complex.

2. Choose one of the complex eigenvalues $\lambda=a+bi$ and construct the usual matrix $C$.

3. Using the same eigenvalue, we will find an eigenvector $\vvec$ where the entri…