# TBIL Activities for Understanding Linear Algebra

## Section4.3Diagonalization, similarity, and powers of a matrix

### Activity4.3.0.1.

Let's recall how a vector in $$\real^2$$ can be represented in a coordinate system defined by a basis $$\bcal=\{\vvec_1, \vvec_2\}\text{.}$$
Suppose that we consider the basis $$\bcal$$ defined by
\begin{equation*} \vvec_1 = \twovec{1}{1},\qquad \vvec_2 = \twovec{-1}{0}\text{.} \end{equation*}

#### (a)

Find the vector $$\xvec$$ whose representation in the coordinate system defined by $$\bcal$$ is $$\coords{\xvec}{\bcal} = \twovec{-3}{2}\text{.}$$

#### (b)

Consider the vector $$\xvec=\twovec{4}{5}$$ and find its representation $$\coords{\xvec}{\bcal}$$ in the coordinate system defined by $$\bcal\text{.}$$

#### (c)

How do we use the matrix $$C_{\bcal} = \left[\begin{array}{rr} \vvec_1 \amp \vvec_2 \end{array}\right]$$ to convert a vector's representation $$\coords{\xvec}{\bcal}$$ in the coordinate system defined by $$\bcal$$ into its standard representation $$\xvec\text{?}$$ How do we use this matrix to convert $$\xvec$$ into $$\coords{\xvec}{\bcal}\text{?}$$

#### (d)

Suppose that we have a matrix $$A$$ whose eigenvectors are $$\vvec_1$$ and $$\vvec_2$$ and associated eigenvalues are $$\lambda_1=4$$ and $$\lambda_2 = 2\text{.}$$ Express the vector $$A(-3\vvec_1 +5\vvec_2)$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (e)

If $$\coords{\xvec}{\bcal} = \twovec{-3}{5}\text{,}$$ find $$\coords{A\xvec}{\bcal}\text{.}$$

### Activity4.3.0.2.

Once again, we will consider the matrices
\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right],\qquad D = \left[\begin{array}{rr} 3 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}
The matrix $$A$$ has eigenvectors $$\vvec_1=\twovec{1}{1}$$ and $$\vvec_2=\twovec{-1}{1}$$ and eigenvalues $$\lambda_1=3$$ and $$\lambda_2=-1\text{.}$$ We will consider the basis of $$\real^2$$ consisting of eigenvectors $$\bcal= \{\vvec_1, \vvec_2\}\text{.}$$

#### (a)

If $$\xvec= 2\vvec_1 - 3\vvec_2\text{,}$$ write $$A\xvec$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (b)

If $$\coords{\xvec}{\bcal}=\twovec{2}{-3}\text{,}$$ find $$\coords{A\xvec}{\bcal}\text{,}$$ the representation of $$A\xvec$$ in the coordinate system defined by $$\bcal\text{.}$$

#### (c)

If $$\coords{\xvec}{\bcal}=\twovec{c_1}{c_2}\text{,}$$ find $$\coords{A\xvec}{\bcal}\text{,}$$ the representation of $$A\xvec$$ in the coordinate system defined by $$\bcal\text{.}$$

#### (d)

Explain why $$\coords{A\xvec}{\bcal} = D\coords{\xvec}{\bcal}\text{.}$$

#### (e)

Explain why $$C_{\bcal}^{-1}A\xvec = DC_{\bcal}^{-1}\xvec$$ for all vectors $$\xvec$$ and hence
\begin{equation*} C_{\bcal}^{-1}A = DC_{\bcal}^{-1}\text{.} \end{equation*}

#### (f)

Explain why $$A = C_{\bcal}DC_{\bcal}^{-1}$$ and verify this relationship by computing $$C_{\bcal}DC_{\bcal}^{-1}$$ in the Sage cell below.
# enter the matrices D and C below
D =
C =
C*D*C.inverse()


### Definition4.3.0.1.

We say that the matrix $$A$$ is diagonalizable if there is a diagonal matrix $$D$$ and invertible matrix $$P$$ such that
\begin{equation*} A = PDP^{-1}\text{.} \end{equation*}

### Activity4.3.0.3.

Find a diagonalization of $$A\text{,}$$ if one exists, when
\begin{equation*} A = \left[\begin{array}{rr} 3 \amp -2 \\ 6 \amp -5 \\ \end{array}\right]\text{.} \end{equation*}

### Activity4.3.0.4.

Can the diagonal matrix
\begin{equation*} A = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -5 \\ \end{array}\right] \end{equation*}
be diagonalized? If so, explain how to find the matrices $$P$$ and $$D\text{.}$$

### Activity4.3.0.5.

Find a diagonalization of $$A\text{,}$$ if one exists, when
\begin{equation*} A = \left[\begin{array}{rrr} -2 \amp 0 \amp 0 \\ 1 \amp -3\amp 0 \\ 2 \amp 0 \amp -3 \\ \end{array}\right]\text{.} \end{equation*}



### Activity4.3.0.6.

Find a diagonalization of $$A\text{,}$$ if one exists, when
\begin{equation*} A = \left[\begin{array}{rrr} -2 \amp 0 \amp 0 \\ 1 \amp -3\amp 0 \\ 2 \amp 1 \amp -3 \\ \end{array}\right]\text{.} \end{equation*}



### Activity4.3.0.7.

Suppose that $$A=PDP^{-1}$$ where
\begin{equation*} D = \left[\begin{array}{rr} 3 \amp 0 \\ 0 \amp -1 \\ \end{array}\right],\qquad P = \left[\begin{array}{cc} \vvec_2 \amp \vvec_1 \end{array}\right] = \left[\begin{array}{rr} 2 \amp 2 \\ 1 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Explain why $$A$$ is invertible.

#### (b)

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

#### (c)

Find a diagonalization of $$A^3\text{.}$$

### Activity4.3.0.8.

Consider the diagonal matrix
\begin{equation*} D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}
Find the powers $$D^2\text{,}$$ $$D^3\text{,}$$ and $$D^4\text{.}$$ What is $$D^k$$ for a general value of $$k\text{?}$$

### Activity4.3.0.9.

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

### Activity4.3.0.10.

Suppose that $$A= PDP^{-1}$$ where
\begin{equation*} D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Remembering that the columns of $$P$$ are eigenvectors of $$A\text{,}$$ explain why $$A^2$$ is diagonalizable and find a diagonalization of $$A^2\text{.}$$

#### (b)

Give another explanation of the diagonalizability of $$A^2$$ by writing
\begin{equation*} A^2 = (PDP^{-1})(PDP^{-1}) = PD(P^{-1}P)DP^{-1}\text{.} \end{equation*}

#### (c)

In the same way, find a diagonalization of $$A^3\text{,}$$ $$A^4\text{,}$$ and $$A^k\text{.}$$

### Activity4.3.0.11.

Suppose that $$A$$ is a diagonalizable $$2\times2$$ matrix with eigenvalues $$\lambda_1 = 0.5$$ and $$\lambda_2=0.1\text{.}$$ What happens to $$A^k$$ as $$k$$ becomes very large?

### Definition4.3.0.2.

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

### Activity4.3.0.12.

In case a matrix $$A$$ has complex eigenvalues, we will find a simpler matrix $$C$$ that is similar to $$A\text{.}$$ In particular, if $$A$$ has an eigenvalue $$\lambda = a+bi\text{,}$$ then $$A$$ is similar to $$C=\left[\begin{array}{rr} a \amp -b \\ b \amp a \\ \end{array}\right] \text{.}$$ We will rewrite $$C$$ in terms of $$r$$ and $$\theta\text{.}$$

#### (a)

Explain why
\begin{equation*} \left[\begin{array}{rr} a \amp -b \\ b \amp a \\ \end{array}\right] = \left[\begin{array}{rr} r\cos\theta \amp -r\sin\theta \\ r\sin\theta \amp r\cos\theta \\ \end{array}\right] = \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right]\text{.} \end{equation*}

#### (b)

Explain why $$C$$ has the geometric effect of rotating vectors by $$\theta$$ and stretching them by a factor of $$r\text{.}$$

#### (c)

Let's now consider the matrix $$A\text{:}$$
\begin{equation*} A = \left[\begin{array}{rr} -2 \amp 2 \\ -5 \amp 4 \\ \end{array}\right] \end{equation*}
whose eigenvalues are $$\lambda_1 = 1+i$$ and $$\lambda_2 = 1-i\text{.}$$ We will choose to focus on one of the eigenvalues $$\lambda_1 = a+bi= 1+i.$$
Form the matrix $$C$$ using these values of $$a$$ and $$b\text{.}$$ Then rewrite the point $$(a,b)$$ in polar coordinates by identifying the values of $$r$$ and $$\theta\text{.}$$ Explain the geometric effect of multiplying vectors of $$C\text{.}$$

#### (d)

Suppose that $$P=\left[\begin{array}{rr} 1 \amp 1 \\ 2 \amp 1 \\ \end{array}\right] \text{.}$$ Verify that $$A = PCP^{-1}\text{.}$$
C =
P =
P*C*P.inverse()


#### (e)

Explain why $$A^k = PC^kP^{-1}\text{.}$$

### Activity4.3.0.13.

In the previous activity, we formed the matrix $$C$$ by choosing the eigenvalue $$\lambda_1=1+i\text{.}$$ Suppose we had instead chosen $$\lambda_2 = 1-i\text{.}$$

#### (a)

Form the matrix $$C'$$

#### (b)

Use polar coordinates to describe the geometric effect of $$C'\text{.}$$

#### (c)

Using the matrix $$P' = \left[\begin{array}{rr} 1 \amp -1 \\ 2 \amp -1 \\ \end{array}\right] \text{,}$$ show that $$A = P'C'P'^{-1}\text{.}$$