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Section 4.3 Diagonalization, similarity, and powers of a matrix
Activity 4.3.1 .
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 4.3.2 .
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 4.3.1 .
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 4.3.3 .
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 4.3.4 .
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 4.3.5 .
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 4.3.6 .
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 4.3.7 .
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 4.3.8 .
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 4.3.9 .
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 4.3.10 .
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 4.3.11 .
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 4.3.2 .
We say that
\(A\) is
similar to
\(B\) if there is an invertible matrix
\(P\) such that
\(A = PBP^{-1}\text{.}\)
Activity 4.3.12 .
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 4.3.13 .
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) (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
\(A = P'C'P'^{-1}\text{.}\)
