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

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 $E_3$.

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 $\cos x$
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 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 $A^{100}$
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 […]

## 1 Response

[…] 1 end{bmatrix} text{ and } begin{bmatrix} -1 \ 1 end{bmatrix},] respectively. (See the post Diagonalize a 2 by 2 Symmetric Matrix for […]