Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors

Problem 423

Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\]
respectively.

Suppose that $A$ is a $2\times 2$ matrix having eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\]
respectively.
Then since $A$ has two distinct eigenvalues, the matrix $A$ is diagonalizable.
As we know eigenvectors, we can diagonalize $A$ by the matrix
\[S:=\begin{bmatrix}
1 & 2\\
0& 1
\end{bmatrix}.\]
That is, we have
\[S^{-1}AS=\begin{bmatrix}
2 & 0\\
0& -1
\end{bmatrix}.\]
The inverse matrix of $S$ is given by
\[S^{-1}=\begin{bmatrix}
1 & -2\\
0& 1
\end{bmatrix}.\]
It follows that we have
\begin{align*}
A&=S\begin{bmatrix}
2 & 0\\
0& -1
\end{bmatrix}S^{-1}\\[6pt]
&=
\begin{bmatrix}
1 & 2\\
0& 1
\end{bmatrix}
\begin{bmatrix}
2 & 0\\
0& -1
\end{bmatrix}
\begin{bmatrix}
1 & -2\\
0& 1
\end{bmatrix}\\[6pt]
&=\begin{bmatrix}
2 & -6\\
0& -1
\end{bmatrix}.
\end{align*}

Therefore, the only matrix satisfying the given conditions is
\[A=\begin{bmatrix}
2 & -6\\
0& -1
\end{bmatrix}.\]

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\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic […]

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Solution.
The answer is No.
Counterexample
We give a counterexample. Consider the $2\times 2$ zero matrix.
The zero matrix is a diagonal matrix, and thus it is diagonalizable.
However, the zero matrix is not […]

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As an example, we solve the following problem.
Diagonalize the matrix
\[A=\begin{bmatrix}
4 & -3 & -3 \\
3 &-2 &-3 \\
-1 & 1 & 2
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by finding a nonsingular […]

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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 […]

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Determine whether the matrix
\[A=\begin{bmatrix}
1 & 4\\
2 & 3
\end{bmatrix}\]
is diagonalizable.
If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
(The Ohio State University, Linear Algebra Final Exam […]

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Definitions/Hint.
Recall the relevant definitions.
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\[A=\begin{bmatrix}
a & b-a\\
0& b
\end{bmatrix}.\]
(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 […]

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\[A=\begin{bmatrix}
0 & 1 & 0 \\
-1 &0 &0 \\
0 & 0 & 2
\end{bmatrix}\]
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 […]