Linear Algebra

Diagonalize a 2 by 2 Symmetric Matrix

Diagonalization Problems and Solutions in Linear Algebra

Problem 629

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 $S^{-1}AS=D$.

Add to solve later

Solution.

The characteristic polynomial $p(t)$ of the matrix $A$ is
\begin{align*}
p(t)&=\det(A-tI)=\begin{vmatrix}
2-t & -1\\
-1& 2-1
\end{vmatrix}\\[6pt] &=(2-t)^2-1 =t^2-4t+3\\
&=(t-1)(t-3).
\end{align*}
It follows that the eigenvalues of $A$ are $\lambda=1, 3$ with algebraic multiplicities are both $1$.
Hence, the geometric multiplicities are $1$ and thus any nonzero vector in eahc eigenspace forms a eigenbasis.

Now let us find a eigenbasis for each eigenspace $E_{\lambda}=\calN(A-\lambda I)$.
For the eigenvalue $1$, we have
\[A-I=\begin{bmatrix}
1 & -1\\
-1& 1
\end{bmatrix}\xrightarrow{R_2+R_1} \begin{bmatrix}
1 & -1\\
0& 0
\end{bmatrix}\] This yields that the eigenvectors corresponding to the eigenvalue $1$ are $x_2\begin{bmatrix}
1 \\
1
\end{bmatrix}$ with $x_2\neq 0$. Hence
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
1
\end{bmatrix} \in E_1\] is an eigenbasis for $E_1$.

Similarly, as we have
\[A-3I=\begin{bmatrix}
-1 & -1\\
-1& -1
\end{bmatrix} \xrightarrow{-R_1}\begin{bmatrix}
1 & 1\\
-1& -1
\end{bmatrix} \xrightarrow{R_2+R_1} \begin{bmatrix}
1 & 1\\
0& 0
\end{bmatrix},\] we see that
\[\mathbf{v}_2=\begin{bmatrix}
-1 \\
1
\end{bmatrix} \in E_3\] is an eigenbasis for $E_3$.

Let
\[S=\begin{bmatrix}
\mathbf{v}_1 & \mathbf{v}_2
\end{bmatrix}=
\begin{bmatrix}
1 & -1\\
1& 1
\end{bmatrix} \text{ and } D=\begin{bmatrix}
1 & 0\\
0& 3
\end{bmatrix}.\]

Then the diagonalization procedure yields that $S$ is nonsingular and $S^{-1}AS= D$.

Add to solve later

More from my site

How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
Diagonalize 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 […]
A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix \[A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}\] is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
$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 […]
If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal Let $A$ and $B$ be $n\times n$ matrices. Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$. Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly […]
$Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix$ Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix \[A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]
Quiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
$Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$$ Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$ 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 […]