Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 466

Let
\[A=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}.\]

(a) Find eigenvalues of the matrix $A$.

(b) Find eigenvectors for each eigenvalue of $A$.

(c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(d) Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.

(e) Calculate $A^{100}$. (You do not have to compute $5^{100}$.)

(f) Calculate
\[(A^3-5A^2+3A+I)^{100}.\] Let $w=2^{100}$. Express the solution in terms of $w$.

 
LoadingAdd to solve later

Solution.

(a) Find eigenvalues of the matrix $A$.

To find the eigenvalues of $A$, we calculate the characteristic polynomial $p(t)$ as follows.
We have
\begin{align*}
p(t)&=\det(A-tI)=\begin{vmatrix}
1-t & 2\\
4& 3-t
\end{vmatrix}\\
&=(1-t)(3-t)-8=t^2-4t-5=(t+1)(t-5).
\end{align*}

The eigenvalues of $A$ are roots of its characteristic polynomial $p(t)$.
Hence the eigenvalues of $A$ are $-1$ and $5$.

(b) Find eigenvectors for each eigenvalue of $A$.

We first determine the eigenvectors of the eigenvalue $-1$ by solving the system $(A+I)\mathbf{x}=\mathbf{0}$.
We have
\begin{align*}
A+I=\begin{bmatrix}
2 & 2\\
4& 4
\end{bmatrix}
\xrightarrow[\text{then } \frac{1}{2}R_1]{R_2-2R_1}
\begin{bmatrix}
1 & 1\\
0& 0
\end{bmatrix}.
\end{align*}
This yields that the eigenvectors corresponding to $-1$ are
\[a\begin{bmatrix}
1 \\
-1
\end{bmatrix}\] for any nonzero scalar $a$.

Next, we find the eigenvectors corresponding to the eigenvalue $5$ by solving $(A-5I)\mathbf{x}=\mathbf{0}$.
We have
\begin{align*}
A-5I=\begin{bmatrix}
-4 & 2\\
4& -2
\end{bmatrix}
\xrightarrow[\text{then } \frac{-1}{4}R_1]{R_2+R_1}
\begin{bmatrix}
1 & -1/2\\
0& 0
\end{bmatrix}.
\end{align*}
It follows that the eigenvectors corresponding to $5$ are
\[a\begin{bmatrix}
1 \\
2
\end{bmatrix}\] for any nonzero scalar $a$.

(c) Diagonalize the matrix $A$.

From part (a) and part (b), we have seen that $A$ has eigenvalues $-1$ and $5$ with corresponding eigenvectors
\[\mathbf{u}=\begin{bmatrix}
1 \\
-1
\end{bmatrix} \text{ and } \mathbf{v}=\begin{bmatrix}
1 \\
2
\end{bmatrix}.\] (Here we chose the scalars $a$ to be $1$ but you could use any nonzero values for the scalars $a$.)

Let
\[S=\begin{bmatrix}
\mathbf{u} & \mathbf{v}
\end{bmatrix}
=
\begin{bmatrix}
1 & 1\\
-1& 2
\end{bmatrix}.\] Then the general procedure of the diagonalization yields that the matrix $S$ is invertible and
\[S^{-1}AS=D,\] where $D$ is the diagonal matrix given by
\[D=\begin{bmatrix}
-1 & 0\\
0& 5
\end{bmatrix}.\]

(d) Diagonalize the matrix $A^3-5A^2+3A+I$.

In part (c), we obtained
\[S^{-1}AS=D,\] where
\[S=\begin{bmatrix}
1 & 1\\
-1& 2
\end{bmatrix} \text{ and } D=\begin{bmatrix}
-1 & 0\\
0& 5
\end{bmatrix}.\]

Note that we have $A=SDS^{-1}$ and
\begin{align*}
A^2&=AA=SDS^{-1}\cdot SDS^{-1}=SD^2S^{-1}\\
A^3&=A^2A=SD^2S^{-1}\cdot SDS^{-1}=SD^3S^{-1}.
\end{align*}

These relations gives
\begin{align*}
A^3-5A^2+3A+I&=SD^3S^{-1}-5SD^2S^{-1}+3SDS^{-1}+I\\
&=S(D^3-5D^2+3D+I)S^{-1}.
\end{align*}

Hence we obtain
\begin{align*}
&S^{-1}(A^3-5A^2+3A+I)S\\
&=D^3-5D^2+3D+I\\
&=\begin{bmatrix}
(-1)^3 & 0\\
0& 5^3
\end{bmatrix}
-5\begin{bmatrix}
(-1)^2 & 0\\
0& 5^2
\end{bmatrix}
+3\begin{bmatrix}
-1 & 0\\
0& 5
\end{bmatrix}
+\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix}\\[6pt] &=\begin{bmatrix}
-8 & 0\\
0& 16
\end{bmatrix}.
\end{align*}
This completes the diagonalization of the matrix $A^3-5A^2+3A+I$.

(e) Calculate $A^{100}$.

In part (d), we have seen that $A=SDS^{-1}$, $A^2=SD^2S^{-1}$, $A^3=SD^3S^{-1}$.
Repeating the same argument (or using mathematical induction), we also have
\[A^{100}=SD^{100}S^{-1}.\]

Thus, we have
\begin{align*}
A^{100}&=SD^{100}S^{-1}\\
&=\begin{bmatrix}
1 & 1\\
-1& 2
\end{bmatrix}
\begin{bmatrix}
-1 & 0\\
0& 5
\end{bmatrix}^{100}
\frac{1}{3}\begin{bmatrix}
2 & -1\\
1& 1
\end{bmatrix}\\[6pt] &=\frac{1}{3}\begin{bmatrix}
1 & 1\\
-1& 2
\end{bmatrix}
\begin{bmatrix}
(-1)^{100} & 0\\
0& 5^{100}
\end{bmatrix}
\begin{bmatrix}
2 & -1\\
1& 1
\end{bmatrix}\\[6pt] &=\frac{1}{3}\begin{bmatrix}
2+5^{100} & -1+5^{100}\\
-2+2\cdot 5^{100}& 1+2\cdot 5^{100}
\end{bmatrix}.
\end{align*}

(f) Calculate $(A^3-5A^2+3A+I)^{100}$.

Let
\[B:=A^3-5A^2+3A+I.\]

In part (d), we obtained
\[S^{-1}BS=\begin{bmatrix}
-8 & 0\\
0& 16
\end{bmatrix}.\] Hence we have $B=S\begin{bmatrix}
-8 & 0\\
0& 16
\end{bmatrix} S^{-1}$, and
\begin{align*}
B^{100}&=S\begin{bmatrix}
-8 & 0\\
0& 16
\end{bmatrix}^{100} S^{-1}\\[6pt] &=S\begin{bmatrix}
(-8)^{100} & 0\\
0& 16^{100}
\end{bmatrix} S^{-1}\\[6pt] &=S\begin{bmatrix}
2^{300} & 0\\
0& 2^{400}
\end{bmatrix} S^{-1}
\\[6pt] &=S\begin{bmatrix}
w^3 & 0\\
0& w^4
\end{bmatrix} S^{-1},
\end{align*}
where we put $w=2^{100}$.

Hence we have
\begin{align*}
B^{100}&=\begin{bmatrix}
1 & 1\\
-1& 2
\end{bmatrix}
\begin{bmatrix}
w^3 & 0\\
0& w^4
\end{bmatrix}
\frac{1}{3}\begin{bmatrix}
2 & -1\\
1& 1
\end{bmatrix}\\[6pt] &=\frac{w^3}{3}\begin{bmatrix}
2+w & -1+w\\
-2+2w& 1-2w
\end{bmatrix}.
\end{align*}
Therefore, the result is
\[(A^3-5A^2+3A+I)^{100}=\frac{w^3}{3}\begin{bmatrix}
2+w & -1+w\\
-2+2w& 1-2w
\end{bmatrix}.\]

More diagonalization problems

More Problems related to the diagonalization of a matrix are gathered in the following page:

Diagonalization of Matrices


LoadingAdd to solve later

Sponsored Links

More from my site

  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix \[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]
  • Diagonalize a 2 by 2 Symmetric MatrixDiagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
  • Diagonalize the Upper Triangular Matrix and Find the Power of the MatrixDiagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
  • Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$ Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$. (a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$. (b) Let \[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix. Using the […]
  • How to Find a Formula of the Power of a MatrixHow to Find a Formula of the Power of a Matrix Let $A= \begin{bmatrix} 1 & 2\\ 2& 1 \end{bmatrix}$. Compute $A^n$ for any $n \in \N$. Plan. We diagonalize the matrix $A$ and use this Problem. Steps. Find eigenvalues and eigenvectors of the matrix $A$. Diagonalize the matrix $A$. Use […]
  • Compute Power of Matrix If Eigenvalues and Eigenvectors Are GivenCompute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […]
  • Find All the Square Roots of a Given 2 by 2 MatrixFind All the Square Roots of a Given 2 by 2 Matrix Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$. Find all the square roots of the matrix \[A=\begin{bmatrix} 2 & 2\\ 2& 2 \end{bmatrix}.\]   Proof. Diagonalize $A$. We first diagonalize the matrix […]

You may also like...

1 Response

  1. 06/21/2017

    […] For a solution of this problem and related questions, see the post “Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$“. […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Problems and solutions in Linear Algebra
Are Linear Transformations of Derivatives and Integrations Linearly Independent?

Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let...

Close