Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue

Ohio State University exam problems and solutions in mathematics

Problem 476

Let
\[A=\begin{bmatrix}
1 & 2 & 1 \\
-1 &4 &1 \\
2 & -4 & 0
\end{bmatrix}.\] The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.

(The Ohio State University, Linear Algebra Final Exam Problem)

 
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Solution.

By definition, the eigenspace $E_2$ corresponding to the eigenvalue $2$ is the null space of the matrix $A-2I$.
That is, we have
\[E_2=\calN(A-2I).\]

We reduce the matrix $A-2I$ by elementary row operations as follows.
\begin{align*}
&A-2I=\begin{bmatrix}
-1 & 2 & 1 \\
-1 &2 &1 \\
2 & -4 & -2
\end{bmatrix}\\[6pt] &\xrightarrow{\substack{R_2-R_1\\R_3+2R_1}}
\begin{bmatrix}
-1 & 2 & 1 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix}
\xrightarrow{-R_1}
\begin{bmatrix}
1 & -2 & -1 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix}.
\end{align*}

Thus, the solutions $\mathbf{x}$ of $(A-2I)\mathbf{x}=\mathbf{0}$ satisfy $x_1=2x_2+x_3$.
Thus, the null space $\calN(A-2I)$ consists of vectors
\[\mathbf{x}=\begin{bmatrix}
2x_2+x_3 \\
x_2 \\
x_3
\end{bmatrix}=x_2\begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}+x_3\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}\] for any scalars $x_2, x_3$.

Hence we have
\begin{align*}
E_2=\calN(A-2I)=\Span\left(\, \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix} \,\right).
\end{align*}

It is straightforward to see that the vectors $\begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}$ are linearly independent, hence they form a basis of $E_2$.

Thus, a basis of $E_2$ is
\[\left\{\, \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix} \,\right\}.\]

Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue (This page)
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Wether Given Subsets in ℝ4 R 4 are Subspaces or Not
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

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