Diagonalization, similarity, and powers of a matrix

Section 4.3 Diagonalization, similarity, and powers of a matrix

Activity 4.3.0.1.

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 \(\coords{A\xvec}{\bcal}\text{.}\)

Activity 4.3.0.2.

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.

Definition 4.3.0.1.

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*}

Activity 4.3.0.3.

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*}

Activity 4.3.0.4.

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 \(D\text{.}\)

Activity 4.3.0.5.

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*}

Activity 4.3.0.6.

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*}

Activity 4.3.0.7.

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 \(A^3\text{.}\)

Activity 4.3.0.8.

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 \(k\text{?}\)

Activity 4.3.0.9.

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 \(\lambda^2\text{.}\)

Activity 4.3.0.10.

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 \(A^k\text{.}\)

Activity 4.3.0.11.

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?

Definition 4.3.0.2.

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

Activity 4.3.0.12.

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 \(A = PCP^{-1}\text{.}\)

(e)

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

Activity 4.3.0.13.

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 \(A = P'C'P'^{-1}\text{.}\)